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

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #121 : Plane Geometry

1

The hypotenuse is the diameter of the circle. Find the area of the circle above.

Possible Answers:

6.5\pi

5\pi

6.25\pi

6.75\pi

5.5\pi

Correct answer:

6.25\pi

Explanation:

Using the Pythagorean Theorem, we can find the length of the hypotenuse:

3^{2}+4^{2}=5^{2}.

Therefore the hypotenuse has length 5.

The area of the circle is \pi r^{2}=\pi \cdot 2.5^{2}=6.25\pi

Example Question #71 : Triangles

Find the length of the hypotenuse.

Triangle_4_14_c

Note: This is a right triangle.

Possible Answers:

Correct answer:

Explanation:

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

, 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:

 

 

 

Example Question #88 : Sat Mathematics

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:

Correct answer:

Explanation:

Use the Pythagorean Theorem: , 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:

Example Question #41 : 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

Example Question #51 : 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 :

Now we can start solving for :

The length of the hypotenuse is .

Example Question #123 : Plane Geometry

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:

Correct answer:

Explanation:

By the Pythagorean Theorem, if  is the hypotenuse and  and  are the legs, .

Set , the known leg, and rewrite the above as:

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

:

 

:

 

:

 

:

 

:

 

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

Example Question #512 : Geometry

Figure6

Find the perimeter of the polygon.

Possible Answers:

Correct answer:

Explanation:

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

Figure7

Find the hypotenuse of the right triangle with the Pythagorean Theorem, , where  and  are the legs of the triangle and  is its hypotenuse. 

This is our missing length.

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

Example Question #513 : 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:

Correct answer:

Explanation:

Since the lengths of the sides are consecutive integers and the shortest side is , the three sides are , , and .

We then use the Pythagorean Theorem:

 

 

Example Question #511 : Psat Mathematics

Square  is on the coordinate plane, and each side of the square is parallel to either the -axis or -axis. Point  has coordinates and point  has the coordinates .

Quantity A:  5\sqrt{2}

Quantity B: The distance between points  and

Possible Answers:

The two quantities are equal.

 

The relationship cannot be determined from the information provided.

 

Quantity B is greater.

 

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 , so if the sides have a length of 5, the hypotenuse must be 5\sqrt{2}.

Example Question #71 : Triangles

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

Possible Answers:

Correct answer:

Explanation:

Plug the values into the Pythagorean Theorem

,

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

.

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