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High School Math : Plane Geometry

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Example Questions

Example Question #46 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Find the length of the hypotenuse.

Triangle_4_14_c

Note: This is a right triangle.

Possible Answers:

$2\sqrt{53}$

$\sqrt{91}$

$4\sqrt{53}$

Correct answer:

$2\sqrt{53}$

Explanation:

To find the length of this hypotenuse, we need to use the Pythagorean Theorem:

c^2=a^2+b^2 , where a and b are the legs and c is the hypotenuse.

Here, c is our missing hypotenuse length, a = 4 ,and b = 14.

Plug these values in and solve for c:

c^2=4^2+14^2=16+196=212

$c=\sqrt{212}=\sqrt{4 \cdot 53}=\sqrt{4}\sqrt{53}=2\sqrt{53}$

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Example Question #47 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Side in the triangle below (not to scale) is equal to . Side is equal to . What is the length of side ?

Right_triangle_with_labeled_sides

Possible Answers:

$\sqrt{146}$

$\sqrt{135}$

$\ 146$

Correct answer:

$\sqrt{146}$

Explanation:

Use the Pythagorean Theorem: $a^{2}+b^{2}=c^{2}$ , where a and b are the legs and c is the hypotenuse.

We know and , so we can plug them in to solve for c:

$5^{2}+11^{2}=c^{2}$

$25+121=c^{2}$

$146=c^{2}$

$c=\sqrt{146}$

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Example Question #48 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Dan drives 5 miles north and then 8 miles west to get to school. If he walks, he can take a direct path from his house to the school, cutting down the distance. How long is the path from Dan's house to his school?

Possible Answers:

13 miles

4.36 miles

19 miles

9.43 miles

89 miles

Correct answer:

9.43 miles

Explanation:

We are really looking for the hypotenuse of a triangle that has legs of 5 miles and 8 miles.

Apply the Pythagorean Theorem:

a² + b² = c²

25 + 64 = c²

89 = c²

c = 9.43 miles

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Example Question #45 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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 #61 : Triangles

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:

$15\ inches$

$4\ inches$

$20\ inches$

$16\ inches$

$12\ inches$

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 #62 : Triangles

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 #63 : Triangles

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)=x

x+x-3=x

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

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

(x)(x+1)=(x+2)^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 #114 : 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:

The relationship cannot be determined from the information provided.

Quantity B is greater.

The two quantities are equal.

Quantity A 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 #194 : Plane Geometry

Justin travels $15\textup{ feet}$ to the east and $20\textup{ feet}$ to the north. How far away from his starting point is he now?

Possible Answers:

$25\textup{ ft}$

$35\textup{ ft}$

$30\textup{ ft}$

$45\textup{ ft}$

$22\textup{ ft}$

Correct answer:

$25\textup{ ft}$

Explanation:

This is solving for the hypotenuse of a triangle. Using the Pythagorean Theorem, which says that a^2+b^2=c^2

15^2 + 20^2 = c^2

225+400=c^2

625=c^2

25=c

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Example Question #195 : Plane Geometry

Susie walks north from her house to a park that is 30 meters away. Once she arrives at the park, she turns and walks west for 80 meters to a bench to feed some pigeons. She then walks north for another 30 meters to a concession stand. If Susie returns home in a straight line from the concession stand, how far will she walk from the concession stand to her house, in meters?

Possible Answers:

200

100

Correct answer:

100

Explanation:

Susie walks 30 meters north, then 80 meters west, then 30 meters north again. Thus, she walks 60 meters north and 80 meters west. These two directions are 90 degrees away from one another.

At this point, construct a right triangle with one leg that measures 60 meters and a second leg that is 80 meters.

You can save time by using the 3:4:5 common triangle. 60 and 80 are $3\cdot 20$ and $4\cdot 20$ , respectively, making the hypotenuse equal to $5\cdot 20=100$ .

We can solve for the length of the missing hypotenuse by applying the Pythagorean theorem:

$a^{2}+b^{2}=c^{2}$

Substitute the following known values into the formula and solve for the missing hypotenuse: side .

$a= 60,\ b= 80,\ c= ?$

$(60)^{2}+(80)^{2}=c^{2}$

$3600+6400=c^{2}$

$10,000=c^{2}$

c=100

Susie will walk 100 meters to reach her house.

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High School Math : Plane Geometry

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #46 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #47 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #48 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #45 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #61 : Triangles

Example Question #62 : Triangles

Example Question #63 : Triangles

Example Question #114 : Geometry

Example Question #194 : Plane Geometry

Example Question #195 : Plane Geometry

All High School Math Resources

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