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ACT Math : How to find the length of the hypotenuse of a right triangle : Pythagorean Theorem

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Example Questions

Example Question #51 : Right Triangles

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 #71 : Right Triangles

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:

$\sqrt{135}$

$\sqrt{146}$

$\ 146$

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 #72 : Right Triangles

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:

89 miles

13 miles

19 miles

4.36 miles

9.43 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 #63 : Right Triangles

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:

$16\ inches$

$12\ inches$

$4\ inches$

$20\ inches$

$15\ inches$

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 #41 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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 #42 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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 #73 : Right Triangles

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 B is greater.

The relationship cannot be determined from the information provided.

Quantity A is greater.

The two quantities are equal.

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 #124 : Geometry

What is the diagonal of a computer screen that measures inches tall by inches wide?

Possible Answers:

18 inches

23 inches

20 inches

7 inches

17 inches

Correct answer:

17 inches

Explanation:

Plug the values into the Pythagorean Theorem

a^2 +b^2 = c^2 ,

because we are solving for the diagonal we are looking for .

8^2 + 15^2 = 289 .

$\sqrt{289} = 17$ .

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Example Question #131 : Geometry

A right triangle has legs of length 20cm and 99 cm , what is the length of the hypotenuse?

Possible Answers:

10,201 cm

119 cm

$\sqrt{119}$

104cm

101 cm

Correct answer:

101 cm

Explanation:

To find the hypotenuse of a right triangle, use the Pythagorean Theorem and plug the leg values in for and :
$\\a^2 + b^2 = c^2 \\ \newline 20^2 + 99^2 = c^2 \\ \newline 10,201 = c^2 \\ \\ \sqrt{10,201}=\sqrt{c^2}\\ \newline 101 = c$

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ACT Math : How to find the length of the hypotenuse of a right triangle : Pythagorean Theorem

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #51 : Right Triangles

Example Question #71 : Right Triangles

Example Question #72 : Right Triangles

Example Question #63 : Right Triangles

Example Question #61 : Triangles

Example Question #41 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #42 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #73 : Right Triangles

Example Question #124 : Geometry

Example Question #131 : Geometry

All ACT Math Resources

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