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Basic Geometry : Triangles

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Example Questions

Example Question #506 : Geometry

What is the hypotenuse of a right triangle with side lengths and ?

Possible Answers:

Correct answer:

Explanation:

The Pythagorean Theorem states that . This question gives us the values of and , and asks us to solve for .

Take and and plug them into the equation as and :

$12^{2}+16^{2}=c^{2}$

Now we can start solving for :

$144+256=c^{2}$

$400=c^{2}$

$\sqrt{400}=\sqrt{c^{2}}$

20=c

The length of the hypotenuse is .

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Example Question #125 : Act Math

One leg of a triangle measures 12 inches. Which of the following could be the length of the other leg if the hypotenuse is an integer length?

Possible Answers:

$16\ inches$

$4\ inches$

$20\ inches$

$15\ inches$

$12\ inches$

Correct answer:

$16\ inches$

Explanation:

By the Pythagorean Theorem, if is the hypotenuse and and are the legs, $c = \sqrt{a^{2}+b^{2}}}$ .

Set a=12 , the known leg, and rewrite the above as:

$c = \sqrt{12^{2}+b^{2}}}$

$c = \sqrt{144+b^{2}}}$

We can now substitute each of the five choices for ; the one which yields a whole number for is the correct answer choice.

b=4 :

$c = \sqrt{144+4^{2}}} = \sqrt{144+16}} =\sqrt{160}=12.64...$

b=12 :

$c = \sqrt{144+12^{2}}} = \sqrt{144+144}} =\sqrt{288}=16.97...$

b=15 :

$c = \sqrt{144+15^{2}}} = \sqrt{144+225}} =\sqrt{369}=19.20...$

b=16 :

$c = \sqrt{144+16^{2}}} = \sqrt{144+256}} =\sqrt{400}=20$

b=20 :

$c = \sqrt{144+20^{2}}} = \sqrt{144+400}} =\sqrt{544}=23.32...$

The only value of which yields a whole number for the hypotenuse is 16, so this is the one we choose.

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Example Question #431 : Geometry

Find the perimeter of the polygon.

Possible Answers:

Correct answer:

Explanation:

Divide the shape into a rectangle and a right triangle as indicated below.

Find the hypotenuse of the right triangle with the Pythagorean Theorem, a^2 + b^2 = c^2 , where and are the legs of the triangle and is its hypotenuse.

(8)^2+(6)^2 = c^2

100 = c^2

$\sqrt{100} = \sqrt{c^2}$

c =10

This is our missing length.

Now add the sides of the polygon together to find the perimeter:

20 + 10 + 26 + 8 = 64

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Example Question #432 : Geometry

The lengths of the sides of a right triangle are consecutive integers, and the length of the shortest side is . Which of the following expressions could be used to solve for ?

Possible Answers:

x^2+(x+1)^2=(x+2)^2

(x)(x+1)=(x+2)^2

x+x-3=x

(x+2)-(x+1)=x

x^2+(x+2)^2=(x+4)^2

Correct answer:

x^2+(x+1)^2=(x+2)^2

Explanation:

Since the lengths of the sides are consecutive integers and the shortest side is , the three sides are , (x+1) , and (x+2) .

We then use the Pythagorean Theorem:

$\newline a^2+b^2=c^2 \newline x^2+(x+1)^2=(x+2)^2$

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Example Question #433 : Geometry

Square PQRS is on the coordinate plane, and each side of the square is parallel to either the -axis or -axis. Point has coordinates $\left ( -2,-1 \right )$ and point has the coordinates $\left ( 3,4 \right )$ .

Quantity A: $5\sqrt{2}$

Quantity B: The distance between points and

Possible Answers:

Quantity A is greater.

The two quantities are equal.

The relationship cannot be determined from the information provided.

Quantity B is greater.

Correct answer:

The two quantities are equal.

Explanation:

To find the distance between points and , split the square into two 45-45-90 triangles and find the hypotenuse. The side ratio of the 45-45-90 triangle is $s:s:s\sqrt{2}$ , so if the sides have a length of 5, the hypotenuse must be $5\sqrt{2}$ .

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Example Question #281 : Triangles

Given the two perpendicular sides of a triangle are and , find the hypotenuse.

Possible Answers:

$\sqrt{58}$

$\sqrt{21}$

Correct answer:

$\sqrt{58}$

Explanation:

To find the length of the hypotenuse, you must use the pythagorean theorem.

c^2=a^2+b^2=3^2+7^2=9+49=58

$c^2=58\Rightarrow c=\sqrt{58}$

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Example Question #52 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Find the length of the hypotenuse of the following right triangle.

Possible Answers:

Correct answer:

Explanation:

Recall the Pythagorean Theorem, which is used to find the length of the hypotenuse.

For any triangle with leg lengths of and ,

$\text{Hypotenuse}^2=a^2+b^2$

Take the square root of both sides to find the length of the hypotenuse.

$\text{Hypotenuse}=\sqrt{a^2+b^2}$

Plug in the given values to find the length of the hypotenuse.

$\text{Hypotenuse}=\sqrt{12^2 + 16^2}=\sqrt{400}=20$

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Example Question #11 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7

Find the length of the hypotenuse of the following right triangle.

Possible Answers:

256

$2\sqrt{111}$

$6\sqrt{13}$

$2\sqrt{113}$

Correct answer:

$2\sqrt{113}$

Explanation:

Recall the Pythagorean Theorem, which is used to find the length of the hypotenuse.

For any triangle with leg lengths of and ,

$\text{Hypotenuse}^2=a^2+b^2$

Take the square root of both sides to find the length of the hypotenuse.

$\text{Hypotenuse}=\sqrt{a^2+b^2}$

Plug in the given values to find the length of the hypotenuse.

$\text{Hypotenuse}=\sqrt{14^2 + 16^2}=\sqrt{452}=2\sqrt{113}$

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Example Question #12 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7

Find the length of the hypotenuse of the following right triangle.

Possible Answers:

$3\sqrt5$

$2\sqrt5$

$\sqrt5$

$4\sqrt5$

Correct answer:

$2\sqrt5$

Explanation:

Recall the Pythagorean Theorem, which is used to find the length of the hypotenuse.

For any triangle with leg lengths of and ,

$\text{Hypotenuse}^2=a^2+b^2$

Take the square root of both sides to find the length of the hypotenuse.

$\text{Hypotenuse}=\sqrt{a^2+b^2}$

Plug in the given values to find the length of the hypotenuse.

$\text{Hypotenuse}=\sqrt{2^2 + 4^2}=\sqrt{20}=2\sqrt5$

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Example Question #12 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7

Find the length of the hypotenuse of the following right triangle.

Possible Answers:

$5\sqrt5$

$4\sqrt5$

$2\sqrt{10}$

$6\sqrt5$

Correct answer:

$5\sqrt5$

Explanation:

Recall the Pythagorean Theorem, which is used to find the length of the hypotenuse.

For any triangle with leg lengths of and ,

$\text{Hypotenuse}^2=a^2+b^2$

Take the square root of both sides to find the length of the hypotenuse.

$\text{Hypotenuse}=\sqrt{a^2+b^2}$

Plug in the given values to find the length of the hypotenuse.

$\text{Hypotenuse}=\sqrt{5^2 + 10^2}=\sqrt{125}=5\sqrt5$

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Basic Geometry : Triangles

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #506 : Geometry

Example Question #125 : Act Math

Example Question #431 : Geometry

Example Question #432 : Geometry

Example Question #433 : Geometry

Example Question #281 : Triangles

Example Question #52 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #11 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7

Example Question #12 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7

Example Question #12 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7

All Basic Geometry Resources

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