Basic Geometry : How to find the perimeter of a right triangle

Study concepts, example questions & explanations for Basic Geometry

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Example Questions

Example Question #21 : How To Find The Perimeter Of A Right Triangle

Find the perimeter.

6

Possible Answers:

\displaystyle 133.36

\displaystyle 132.09

\displaystyle 134.51

\displaystyle 125.18

Correct answer:

\displaystyle 134.51

Explanation:

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

  1. When side lengths are given, add them together.
  2. Solve for a missing side using the Pythagorean theorem.
  3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.

 

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

Screen shot 2016 07 07 at 11.31.44 am

If we know the lengths of sides \displaystyle a\displaystyle b, and \displaystyle c, then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

 

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one \displaystyle 90^{\circ} angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the \displaystyle 90^{\circ} angle and they are labeled \displaystyle a and \displaystyle b. The side of the triangle that is opposite of the \displaystyle 90^{\circ} angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as \displaystyle c.

 

Screen shot 2016 07 07 at 10.13.54 am

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

\displaystyle a^2+b^2=c^2

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, \displaystyle c.

\displaystyle a^2+b^2=c^2

Rearrange and take the square root of both sides. 

\displaystyle \sqrt{c^2}=\sqrt{a^2+b^2}

Simplify.

\displaystyle c=\sqrt{a^2+b^2}

Now, let's use the Pythagorean theorem to solve for one of the legs, \displaystyle a.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle b^2 from both sides of the equation.

\displaystyle a^2+b^2-b^2=c^2-b^2

\displaystyle a^2=c^2-b^2

Take the square root of both sides.

\displaystyle \sqrt{a^2}=\sqrt{c^2-b^2}

Simplify.

\displaystyle a=\sqrt{c^2-b^2}

Last, let's use the Pythagorean theorem to solve for the adjacent leg, \displaystyle b.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle a^2 from both sides of the equation.

\displaystyle a^2-a^2+b^2=c^2-a^2

\displaystyle b^2=c^2-a^2

Take the square root of both sides.

\displaystyle \sqrt{b^2}=\sqrt{c^2-a^2}

Simplify.

\displaystyle b=\sqrt{c^2-a^2}

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

\displaystyle a=\sqrt{c^2-b^2}

\displaystyle b=\sqrt{c^2-a^2}

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.  

 

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

Screen shot 2016 07 07 at 12.58.14 pm

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

\displaystyle a^2=b^2+c^2-2bc\cdot \cos A

\displaystyle b^2=a^2+c^2-2ac\cdot \cos B

\displaystyle c^2=a^2+b^2-2ab\cdot \cos C

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

 

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

13

In order to find the perimeter, we must first find the length of the hypotenuse of the right triangle.

Use Pythagorean's theorem to find the length of the hypotenuse:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

Plug in the values of the lengths of the legs of the given triangle.

\displaystyle c=\sqrt{8^2+63^2}=\sqrt{4033}

Now, recall how to find the perimeter of a triangle:

\displaystyle \text{Perimeter}=a+b+c

Plug in all the values of the sides of the triangle to find the perimeter.

\displaystyle \text{Perimeter}=8+63+\sqrt{4033}=134.51

Make sure to round to \displaystyle 2 places after the decimal.

Example Question #21 : How To Find The Perimeter Of A Right Triangle

Find the perimeter.

7

Possible Answers:

\displaystyle 70.11

\displaystyle 74.21

\displaystyle 73.81

\displaystyle 76.91

Correct answer:

\displaystyle 73.81

Explanation:

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

  1. When side lengths are given, add them together.
  2. Solve for a missing side using the Pythagorean theorem.
  3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.

 

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

Screen shot 2016 07 07 at 11.31.44 am

If we know the lengths of sides \displaystyle a\displaystyle b, and \displaystyle c, then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

 

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one \displaystyle 90^{\circ} angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the \displaystyle 90^{\circ} angle and they are labeled \displaystyle a and \displaystyle b. The side of the triangle that is opposite of the \displaystyle 90^{\circ} angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as \displaystyle c.

 

Screen shot 2016 07 07 at 10.13.54 am

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

\displaystyle a^2+b^2=c^2

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, \displaystyle c.

\displaystyle a^2+b^2=c^2

Rearrange and take the square root of both sides. 

\displaystyle \sqrt{c^2}=\sqrt{a^2+b^2}

Simplify.

\displaystyle c=\sqrt{a^2+b^2}

Now, let's use the Pythagorean theorem to solve for one of the legs, \displaystyle a.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle b^2 from both sides of the equation.

\displaystyle a^2+b^2-b^2=c^2-b^2

\displaystyle a^2=c^2-b^2

Take the square root of both sides.

\displaystyle \sqrt{a^2}=\sqrt{c^2-b^2}

Simplify.

\displaystyle a=\sqrt{c^2-b^2}

Last, let's use the Pythagorean theorem to solve for the adjacent leg, \displaystyle b.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle a^2 from both sides of the equation.

\displaystyle a^2-a^2+b^2=c^2-a^2

\displaystyle b^2=c^2-a^2

Take the square root of both sides.

\displaystyle \sqrt{b^2}=\sqrt{c^2-a^2}

Simplify.

\displaystyle b=\sqrt{c^2-a^2}

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

\displaystyle a=\sqrt{c^2-b^2}

\displaystyle b=\sqrt{c^2-a^2}

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.  

 

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

Screen shot 2016 07 07 at 12.58.14 pm

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

\displaystyle a^2=b^2+c^2-2bc\cdot \cos A

\displaystyle b^2=a^2+c^2-2ac\cdot \cos B

\displaystyle c^2=a^2+b^2-2ab\cdot \cos C

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

 

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

13

In order to find the perimeter, we must first find the length of the hypotenuse of the right triangle.

Use Pythagorean's theorem to find the length of the hypotenuse:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

Plug in the values of the lengths of the legs of the given triangle.

\displaystyle c=\sqrt{18^2+25^2}=\sqrt{949}

Now, recall how to find the perimeter of a triangle:

\displaystyle \text{Perimeter}=a+b+c

Plug in all the values of the sides of the triangle to find the perimeter.

\displaystyle \text{Perimeter}=18+25+\sqrt{949}=73.81

Make sure to round to \displaystyle 2 places after the decimal.

Example Question #368 : Triangles

Find the perimeter.

8

Possible Answers:

\displaystyle 94.67

\displaystyle 102.24

\displaystyle 90.26

\displaystyle 95.06

Correct answer:

\displaystyle 95.06

Explanation:

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

  1. When side lengths are given, add them together.
  2. Solve for a missing side using the Pythagorean theorem.
  3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.

 

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

Screen shot 2016 07 07 at 11.31.44 am

If we know the lengths of sides \displaystyle a\displaystyle b, and \displaystyle c, then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

 

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one \displaystyle 90^{\circ} angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the \displaystyle 90^{\circ} angle and they are labeled \displaystyle a and \displaystyle b. The side of the triangle that is opposite of the \displaystyle 90^{\circ} angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as \displaystyle c.

 

Screen shot 2016 07 07 at 10.13.54 am

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

\displaystyle a^2+b^2=c^2

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, \displaystyle c.

\displaystyle a^2+b^2=c^2

Rearrange and take the square root of both sides. 

\displaystyle \sqrt{c^2}=\sqrt{a^2+b^2}

Simplify.

\displaystyle c=\sqrt{a^2+b^2}

Now, let's use the Pythagorean theorem to solve for one of the legs, \displaystyle a.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle b^2 from both sides of the equation.

\displaystyle a^2+b^2-b^2=c^2-b^2

\displaystyle a^2=c^2-b^2

Take the square root of both sides.

\displaystyle \sqrt{a^2}=\sqrt{c^2-b^2}

Simplify.

\displaystyle a=\sqrt{c^2-b^2}

Last, let's use the Pythagorean theorem to solve for the adjacent leg, \displaystyle b.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle a^2 from both sides of the equation.

\displaystyle a^2-a^2+b^2=c^2-a^2

\displaystyle b^2=c^2-a^2

Take the square root of both sides.

\displaystyle \sqrt{b^2}=\sqrt{c^2-a^2}

Simplify.

\displaystyle b=\sqrt{c^2-a^2}

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

\displaystyle a=\sqrt{c^2-b^2}

\displaystyle b=\sqrt{c^2-a^2}

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.  

 

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

Screen shot 2016 07 07 at 12.58.14 pm

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

\displaystyle a^2=b^2+c^2-2bc\cdot \cos A

\displaystyle b^2=a^2+c^2-2ac\cdot \cos B

\displaystyle c^2=a^2+b^2-2ab\cdot \cos C

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

 

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

13

In order to find the perimeter, we must first find the length of the hypotenuse of the right triangle.

Use Pythagorean's theorem to find the length of the hypotenuse:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

Plug in the values of the lengths of the legs of the given triangle.

\displaystyle c=\sqrt{13^2+40^2}=\sqrt{1769}

Now, recall how to find the perimeter of a triangle:

\displaystyle \text{Perimeter}=a+b+c

Plug in all the values of the sides of the triangle to find the perimeter.

\displaystyle \text{Perimeter}=13+40+\sqrt{1769}=95.06

Make sure to round to \displaystyle 2 places after the decimal.

Example Question #369 : Triangles

Find the perimeter.

9

Possible Answers:

\displaystyle 63.91

\displaystyle 64.73

\displaystyle 62.09

\displaystyle 62.33

Correct answer:

\displaystyle 64.73

Explanation:

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

  1. When side lengths are given, add them together.
  2. Solve for a missing side using the Pythagorean theorem.
  3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.

 

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

Screen shot 2016 07 07 at 11.31.44 am

If we know the lengths of sides \displaystyle a\displaystyle b, and \displaystyle c, then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

 

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one \displaystyle 90^{\circ} angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the \displaystyle 90^{\circ} angle and they are labeled \displaystyle a and \displaystyle b. The side of the triangle that is opposite of the \displaystyle 90^{\circ} angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as \displaystyle c.

 

Screen shot 2016 07 07 at 10.13.54 am

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

\displaystyle a^2+b^2=c^2

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, \displaystyle c.

\displaystyle a^2+b^2=c^2

Rearrange and take the square root of both sides. 

\displaystyle \sqrt{c^2}=\sqrt{a^2+b^2}

Simplify.

\displaystyle c=\sqrt{a^2+b^2}

Now, let's use the Pythagorean theorem to solve for one of the legs, \displaystyle a.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle b^2 from both sides of the equation.

\displaystyle a^2+b^2-b^2=c^2-b^2

\displaystyle a^2=c^2-b^2

Take the square root of both sides.

\displaystyle \sqrt{a^2}=\sqrt{c^2-b^2}

Simplify.

\displaystyle a=\sqrt{c^2-b^2}

Last, let's use the Pythagorean theorem to solve for the adjacent leg, \displaystyle b.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle a^2 from both sides of the equation.

\displaystyle a^2-a^2+b^2=c^2-a^2

\displaystyle b^2=c^2-a^2

Take the square root of both sides.

\displaystyle \sqrt{b^2}=\sqrt{c^2-a^2}

Simplify.

\displaystyle b=\sqrt{c^2-a^2}

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

\displaystyle a=\sqrt{c^2-b^2}

\displaystyle b=\sqrt{c^2-a^2}

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.  

 

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

Screen shot 2016 07 07 at 12.58.14 pm

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

\displaystyle a^2=b^2+c^2-2bc\cdot \cos A

\displaystyle b^2=a^2+c^2-2ac\cdot \cos B

\displaystyle c^2=a^2+b^2-2ab\cdot \cos C

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

 

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

13

In order to find the perimeter, we must first find the length of the hypotenuse of the right triangle.

Use Pythagorean's theorem to find the length of the hypotenuse:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

Plug in the values of the lengths of the legs of the given triangle.

\displaystyle c=\sqrt{12^2+25^2}=\sqrt{769}

Now, recall how to find the perimeter of a triangle:

\displaystyle \text{Perimeter}=a+b+c

Plug in all the values of the sides of the triangle to find the perimeter.

\displaystyle \text{Perimeter}=12+25+\sqrt{769}=64.73

Make sure to round to \displaystyle 2 places after the decimal.

Example Question #370 : Triangles

Find the perimeter.

10

Possible Answers:

\displaystyle 76.11

\displaystyle 72.58

\displaystyle 75.90

\displaystyle 72.95

Correct answer:

\displaystyle 72.95

Explanation:

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

  1. When side lengths are given, add them together.
  2. Solve for a missing side using the Pythagorean theorem.
  3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.

 

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

Screen shot 2016 07 07 at 11.31.44 am

If we know the lengths of sides \displaystyle a\displaystyle b, and \displaystyle c, then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

 

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one \displaystyle 90^{\circ} angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the \displaystyle 90^{\circ} angle and they are labeled \displaystyle a and \displaystyle b. The side of the triangle that is opposite of the \displaystyle 90^{\circ} angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as \displaystyle c.

 

Screen shot 2016 07 07 at 10.13.54 am

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

\displaystyle a^2+b^2=c^2

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, \displaystyle c.

\displaystyle a^2+b^2=c^2

Rearrange and take the square root of both sides. 

\displaystyle \sqrt{c^2}=\sqrt{a^2+b^2}

Simplify.

\displaystyle c=\sqrt{a^2+b^2}

Now, let's use the Pythagorean theorem to solve for one of the legs, \displaystyle a.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle b^2 from both sides of the equation.

\displaystyle a^2+b^2-b^2=c^2-b^2

\displaystyle a^2=c^2-b^2

Take the square root of both sides.

\displaystyle \sqrt{a^2}=\sqrt{c^2-b^2}

Simplify.

\displaystyle a=\sqrt{c^2-b^2}

Last, let's use the Pythagorean theorem to solve for the adjacent leg, \displaystyle b.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle a^2 from both sides of the equation.

\displaystyle a^2-a^2+b^2=c^2-a^2

\displaystyle b^2=c^2-a^2

Take the square root of both sides.

\displaystyle \sqrt{b^2}=\sqrt{c^2-a^2}

Simplify.

\displaystyle b=\sqrt{c^2-a^2}

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

\displaystyle a=\sqrt{c^2-b^2}

\displaystyle b=\sqrt{c^2-a^2}

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.  

 

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

Screen shot 2016 07 07 at 12.58.14 pm

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

\displaystyle a^2=b^2+c^2-2bc\cdot \cos A

\displaystyle b^2=a^2+c^2-2ac\cdot \cos B

\displaystyle c^2=a^2+b^2-2ab\cdot \cos C

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

 

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

13

In order to find the perimeter, we must first find the length of the hypotenuse of the right triangle.

Use Pythagorean's theorem to find the length of the hypotenuse:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

Plug in the values of the lengths of the legs of the given triangle.

\displaystyle c=\sqrt{11^2+30^2}=\sqrt{1021}

Now, recall how to find the perimeter of a triangle:

\displaystyle \text{Perimeter}=a+b+c

Plug in all the values of the sides of the triangle to find the perimeter.

\displaystyle \text{Perimeter}=11+30+\sqrt{1021}=72.95

Make sure to round to \displaystyle 2 places after the decimal.

Example Question #371 : Triangles

Find the perimeter.

12

Possible Answers:

\displaystyle 34.52

\displaystyle 28.57

\displaystyle 33.28

\displaystyle 31.09

Correct answer:

\displaystyle 34.52

Explanation:

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

  1. When side lengths are given, add them together.
  2. Solve for a missing side using the Pythagorean theorem.
  3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.

 

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

Screen shot 2016 07 07 at 11.31.44 am

If we know the lengths of sides \displaystyle a\displaystyle b, and \displaystyle c, then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

 

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one \displaystyle 90^{\circ} angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the \displaystyle 90^{\circ} angle and they are labeled \displaystyle a and \displaystyle b. The side of the triangle that is opposite of the \displaystyle 90^{\circ} angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as \displaystyle c.

 

Screen shot 2016 07 07 at 10.13.54 am

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

\displaystyle a^2+b^2=c^2

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, \displaystyle c.

\displaystyle a^2+b^2=c^2

Rearrange and take the square root of both sides. 

\displaystyle \sqrt{c^2}=\sqrt{a^2+b^2}

Simplify.

\displaystyle c=\sqrt{a^2+b^2}

Now, let's use the Pythagorean theorem to solve for one of the legs, \displaystyle a.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle b^2 from both sides of the equation.

\displaystyle a^2+b^2-b^2=c^2-b^2

\displaystyle a^2=c^2-b^2

Take the square root of both sides.

\displaystyle \sqrt{a^2}=\sqrt{c^2-b^2}

Simplify.

\displaystyle a=\sqrt{c^2-b^2}

Last, let's use the Pythagorean theorem to solve for the adjacent leg, \displaystyle b.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle a^2 from both sides of the equation.

\displaystyle a^2-a^2+b^2=c^2-a^2

\displaystyle b^2=c^2-a^2

Take the square root of both sides.

\displaystyle \sqrt{b^2}=\sqrt{c^2-a^2}

Simplify.

\displaystyle b=\sqrt{c^2-a^2}

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

\displaystyle a=\sqrt{c^2-b^2}

\displaystyle b=\sqrt{c^2-a^2}

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.  

 

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

Screen shot 2016 07 07 at 12.58.14 pm

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

\displaystyle a^2=b^2+c^2-2bc\cdot \cos A

\displaystyle b^2=a^2+c^2-2ac\cdot \cos B

\displaystyle c^2=a^2+b^2-2ab\cdot \cos C

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

 

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

13

In order to find the perimeter, we must first find the length of the hypotenuse of the right triangle.

Use Pythagorean's theorem to find the length of the hypotenuse:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

Plug in the values of the lengths of the legs of the given triangle.

\displaystyle c=\sqrt{4^2+15^2}=\sqrt{241}

Now, recall how to find the perimeter of a triangle:

\displaystyle \text{Perimeter}=a+b+c

Plug in all the values of the sides of the triangle to find the perimeter.

\displaystyle \text{Perimeter}=4+15+\sqrt{241}=34.52

Make sure to round to \displaystyle 2 places after the decimal.

Example Question #372 : Triangles

Find the perimeter.

11

Possible Answers:

\displaystyle 27.37

\displaystyle 20.56

\displaystyle 19.82

\displaystyle 28.15

Correct answer:

\displaystyle 27.37

Explanation:

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

  1. When side lengths are given, add them together.
  2. Solve for a missing side using the Pythagorean theorem.
  3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.

 

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

Screen shot 2016 07 07 at 11.31.44 am

If we know the lengths of sides \displaystyle a\displaystyle b, and \displaystyle c, then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

 

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one \displaystyle 90^{\circ} angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the \displaystyle 90^{\circ} angle and they are labeled \displaystyle a and \displaystyle b. The side of the triangle that is opposite of the \displaystyle 90^{\circ} angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as \displaystyle c.

 

Screen shot 2016 07 07 at 10.13.54 am

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

\displaystyle a^2+b^2=c^2

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, \displaystyle c.

\displaystyle a^2+b^2=c^2

Rearrange and take the square root of both sides. 

\displaystyle \sqrt{c^2}=\sqrt{a^2+b^2}

Simplify.

\displaystyle c=\sqrt{a^2+b^2}

Now, let's use the Pythagorean theorem to solve for one of the legs, \displaystyle a.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle b^2 from both sides of the equation.

\displaystyle a^2+b^2-b^2=c^2-b^2

\displaystyle a^2=c^2-b^2

Take the square root of both sides.

\displaystyle \sqrt{a^2}=\sqrt{c^2-b^2}

Simplify.

\displaystyle a=\sqrt{c^2-b^2}

Last, let's use the Pythagorean theorem to solve for the adjacent leg, \displaystyle b.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle a^2 from both sides of the equation.

\displaystyle a^2-a^2+b^2=c^2-a^2

\displaystyle b^2=c^2-a^2

Take the square root of both sides.

\displaystyle \sqrt{b^2}=\sqrt{c^2-a^2}

Simplify.

\displaystyle b=\sqrt{c^2-a^2}

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

\displaystyle a=\sqrt{c^2-b^2}

\displaystyle b=\sqrt{c^2-a^2}

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.  

 

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

Screen shot 2016 07 07 at 12.58.14 pm

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

\displaystyle a^2=b^2+c^2-2bc\cdot \cos A

\displaystyle b^2=a^2+c^2-2ac\cdot \cos B

\displaystyle c^2=a^2+b^2-2ab\cdot \cos C

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

 

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

13

In order to find the perimeter, we must first find the length of the hypotenuse of the right triangle.

Use Pythagorean's theorem to find the length of the hypotenuse:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

Plug in the values of the lengths of the legs of the given triangle.

\displaystyle c=\sqrt{3^2+12^2}=\sqrt{153}

Now, recall how to find the perimeter of a triangle:

\displaystyle \text{Perimeter}=a+b+c

Plug in all the values of the sides of the triangle to find the perimeter.

\displaystyle \text{Perimeter}=3+12+\sqrt{153}=27.37

Make sure to round to \displaystyle 2 places after the decimal.

Example Question #373 : Triangles

Find the perimeter of a right isosceles triangle with the following side length:

Possible Answers:

\displaystyle 36cm

\displaystyle 18cm

\displaystyle 84cm

\displaystyle 12cm

\displaystyle 20.49cm

Correct answer:

\displaystyle 20.49cm

Explanation:

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

  1. When side lengths are given, add them together.
  2. Solve for a missing side using the Pythagorean theorem.
  3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.

 

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

Screen shot 2016 07 07 at 11.31.44 am

If we know the lengths of sides \displaystyle a\displaystyle b, and \displaystyle c, then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

 

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one \displaystyle 90^{\circ} angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the \displaystyle 90^{\circ} angle and they are labeled \displaystyle a and \displaystyle b. The side of the triangle that is opposite of the \displaystyle 90^{\circ} angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as \displaystyle c.

 

Screen shot 2016 07 07 at 10.13.54 am

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

\displaystyle a^2+b^2=c^2

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, \displaystyle c.

\displaystyle a^2+b^2=c^2

Rearrange and take the square root of both sides. 

\displaystyle \sqrt{c^2}=\sqrt{a^2+b^2}

Simplify.

\displaystyle c=\sqrt{a^2+b^2}

Now, let's use the Pythagorean theorem to solve for one of the legs, \displaystyle a.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle b^2 from both sides of the equation.

\displaystyle a^2+b^2-b^2=c^2-b^2

\displaystyle a^2=c^2-b^2

Take the square root of both sides.

\displaystyle \sqrt{a^2}=\sqrt{c^2-b^2}

Simplify.

\displaystyle a=\sqrt{c^2-b^2}

Last, let's use the Pythagorean theorem to solve for the adjacent leg, \displaystyle b.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle a^2 from both sides of the equation.

\displaystyle a^2-a^2+b^2=c^2-a^2

\displaystyle b^2=c^2-a^2

Take the square root of both sides.

\displaystyle \sqrt{b^2}=\sqrt{c^2-a^2}

Simplify.

\displaystyle b=\sqrt{c^2-a^2}

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

\displaystyle a=\sqrt{c^2-b^2}

\displaystyle b=\sqrt{c^2-a^2}

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.  

 

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

Screen shot 2016 07 07 at 12.58.14 pm

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

\displaystyle a^2=b^2+c^2-2bc\cdot \cos A

\displaystyle b^2=a^2+c^2-2ac\cdot \cos B

\displaystyle c^2=a^2+b^2-2ab\cdot \cos C

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

 

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

Recall how to find the perimeter of a triangle:

\displaystyle \text{Perimeter}=a+b+c

The given triangle has \displaystyle 2 of the three sides needed. Use the Pythagorean theorem to find the length of the third side.

Recall the Pythagorean theorem:

\displaystyle a^2+b^2=c^2

In this case, perimeter is side+side+hypotenuse. The two sides are 6cm, we need to find the hypotenuse using Pythagorean theorem.

\displaystyle a^2+b^2=c^2

\displaystyle 6^2+6^2=c^2

\displaystyle c^2=72

\displaystyle c=8.49

\displaystyle Perimeter = 6+6+8.49=20.49cm

Example Question #374 : Triangles

Find the perimeter, \displaystyle P, of a right triangle whose sides are A,B, and hypotenuse C.

Given, \displaystyle A=7, B=24

Possible Answers:

\displaystyle P=52

\displaystyle P=54

\displaystyle P=50

\displaystyle P=56

Correct answer:

\displaystyle P=56

Explanation:

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

  1. When side lengths are given, add them together.
  2. Solve for a missing side using the Pythagorean theorem.
  3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.

 

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

Screen shot 2016 07 07 at 11.31.44 am

If we know the lengths of sides \displaystyle a\displaystyle b, and \displaystyle c, then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

 

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one \displaystyle 90^{\circ} angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the \displaystyle 90^{\circ} angle and they are labeled \displaystyle a and \displaystyle b. The side of the triangle that is opposite of the \displaystyle 90^{\circ} angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as \displaystyle c.

 

Screen shot 2016 07 07 at 10.13.54 am

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

\displaystyle a^2+b^2=c^2

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, \displaystyle c.

\displaystyle a^2+b^2=c^2

Rearrange and take the square root of both sides. 

\displaystyle \sqrt{c^2}=\sqrt{a^2+b^2}

Simplify.

\displaystyle c=\sqrt{a^2+b^2}

Now, let's use the Pythagorean theorem to solve for one of the legs, \displaystyle a.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle b^2 from both sides of the equation.

\displaystyle a^2+b^2-b^2=c^2-b^2

\displaystyle a^2=c^2-b^2

Take the square root of both sides.

\displaystyle \sqrt{a^2}=\sqrt{c^2-b^2}

Simplify.

\displaystyle a=\sqrt{c^2-b^2}

Last, let's use the Pythagorean theorem to solve for the adjacent leg, \displaystyle b.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle a^2 from both sides of the equation.

\displaystyle a^2-a^2+b^2=c^2-a^2

\displaystyle b^2=c^2-a^2

Take the square root of both sides.

\displaystyle \sqrt{b^2}=\sqrt{c^2-a^2}

Simplify.

\displaystyle b=\sqrt{c^2-a^2}

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

\displaystyle a=\sqrt{c^2-b^2}

\displaystyle b=\sqrt{c^2-a^2}

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.  

 

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

Screen shot 2016 07 07 at 12.58.14 pm

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

\displaystyle a^2=b^2+c^2-2bc\cdot \cos A

\displaystyle b^2=a^2+c^2-2ac\cdot \cos B

\displaystyle c^2=a^2+b^2-2ab\cdot \cos C

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

 

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

Recall how to find the perimeter of a triangle:

\displaystyle \text{Perimeter}=a+b+c

The given triangle has \displaystyle 2 of the three sides needed. Use the Pythagorean theorem to find the length of the third side.

Recall the Pythagorean theorem:

\displaystyle a^2+b^2=c^2

Rearrange to solve for the hypotenuse.

\displaystyle c=\sqrt{a^2+b^2}

\displaystyle C=\sqrt{49+576}=\sqrt{625}=25

We then add up all of the sides. 

Example Question #181 : Right Triangles

Which of the following could NOT be the perimeter of a right triangle with sides of length \displaystyle 4 and \displaystyle 7

Possible Answers:

All of these work 

\displaystyle 11 + \sqrt{65}

None of these work

\displaystyle 11 + \sqrt{33}

\displaystyle 11 + \sqrt{21}

Correct answer:

\displaystyle 11 + \sqrt{21}

Explanation:

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

  1. When side lengths are given, add them together.
  2. Solve for a missing side using the Pythagorean theorem.
  3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.

 

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

Screen shot 2016 07 07 at 11.31.44 am

If we know the lengths of sides \displaystyle a\displaystyle b, and \displaystyle c, then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

 

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one \displaystyle 90^{\circ} angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the \displaystyle 90^{\circ} angle and they are labeled \displaystyle a and \displaystyle b. The side of the triangle that is opposite of the \displaystyle 90^{\circ} angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as \displaystyle c.

 

Screen shot 2016 07 07 at 10.13.54 am

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

\displaystyle a^2+b^2=c^2

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, \displaystyle c.

\displaystyle a^2+b^2=c^2

Rearrange and take the square root of both sides. 

\displaystyle \sqrt{c^2}=\sqrt{a^2+b^2}

Simplify.

\displaystyle c=\sqrt{a^2+b^2}

Now, let's use the Pythagorean theorem to solve for one of the legs, \displaystyle a.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle b^2 from both sides of the equation.

\displaystyle a^2+b^2-b^2=c^2-b^2

\displaystyle a^2=c^2-b^2

Take the square root of both sides.

\displaystyle \sqrt{a^2}=\sqrt{c^2-b^2}

Simplify.

\displaystyle a=\sqrt{c^2-b^2}

Last, let's use the Pythagorean theorem to solve for the adjacent leg, \displaystyle b.

\displaystyle a^2+b^2=c^2

Subtract \displaystyle a^2 from both sides of the equation.

\displaystyle a^2-a^2+b^2=c^2-a^2

\displaystyle b^2=c^2-a^2

Take the square root of both sides.

\displaystyle \sqrt{b^2}=\sqrt{c^2-a^2}

Simplify.

\displaystyle b=\sqrt{c^2-a^2}

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

\displaystyle a^2+b^2=c^2

\displaystyle c=\sqrt{a^2+b^2}

\displaystyle a=\sqrt{c^2-b^2}

\displaystyle b=\sqrt{c^2-a^2}

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.  

 

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

Screen shot 2016 07 07 at 12.58.14 pm

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

\displaystyle a^2=b^2+c^2-2bc\cdot \cos A

\displaystyle b^2=a^2+c^2-2ac\cdot \cos B

\displaystyle c^2=a^2+b^2-2ab\cdot \cos C

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

 

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

Since it wasn't specified if the two lengths given were the legs, or a leg and the hypotenuse, there are 2 options for the third side length. 

Option 1: 

\displaystyle 7^2 + 4^2 = c^2

\displaystyle 49+ 16 = c^2

\displaystyle 65 = c^2

\displaystyle \sqrt{65 } = c

To find the perimeter, add the following:

\displaystyle 4 + 7 + \sqrt{65} = 11 + \sqrt{65}

Option 2: 

\displaystyle 7^2 = 4^2 + b^2

\displaystyle 49 = 16 + b^2

\displaystyle 33 = b^2

\displaystyle \sqrt{33 } = b

To find the perimeter, add the following:

\displaystyle 4+ 7 + \sqrt{33} = 11 + \sqrt{33}

The only option listed that isn't either of those is \displaystyle 11 + \sqrt{21}, so that is the answer. 

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