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

## Example Questions

### Example Question #502 : Geometry

Find the length of the hypotenuse.

Note: This is a right triangle.

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 #41 : 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 ?

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 #61 : 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?

89 miles

19 miles

9.43 miles

13 miles

4.36 miles

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 #41 : 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 ?

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 #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?

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

Find the perimeter of the polygon.

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, , 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 #82 : Right 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 ?

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 #121 : Act Math

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:

Quantity B: The distance between points  and

The two quantities are equal.

The relationship cannot be determined from the information provided.

Quantity B is greater.

Quantity A is greater.

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 .

### Example Question #61 : Triangles

Two sides of a given triangle are both . If one angle of the triangle is a right angle, then what is the measure of the hypotenuse?

Explanation:

If we know two sides are equal to  and we know that one of the angles is a right angle, then that means that this must be a Special Right Triangle where the interior angles are

With this special triangle, we also know that the measure of the hypotenuse is equal to the measure of one side of the triangle times the square root of that measure.

Since one leg of the triangle is . then the hypotenuse is equal to .

We could also solve this using the Pythagorean Theorem, like so:

### Example Question #51 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Justin travels  to the east and  to the north. How far away from his starting point is he now?