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

## Example Questions

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

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

Explanation:

To find the hypotenuse of a right triangle, use the Pythagorean Theorem and plug the leg values in for  and :

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

A right triangle has a base of six and a height of eight. Using this information find the hypotenuse.

Explanation:

This question calls for us to use the Pythagorean Theorem. This theorem has a formula of

where a and b are the sides of a right triangle, adjacent to the right angle, and c is the hypotenuse.

Given our information

.

### Example Question #53 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

A right triangle has a base of seven and a height of twelve. Using this information find the hypotenuse.

Explanation:

This question calls for us to use the Pythagorean Theorem. This theorem has a formula of

where a and b are the sides of a right triangle, adjacent to the right angle, and c is the hypotenuse.

Given our information

.

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

Marcus Absent is marking out some lines for a large canvas tent. He paces out  to the north, places a peg, then turns east and paces another  to place another peg. Stopping, he realizes he has forgotten the other pegs, so he makes a beeline for his original starting point.

How many  does Marcus travel to get back to the pegs?

Explanation:

The Pythagorean Theorem states that for any right triangle with legs  and  and hypotenuse :

.

Applying this to Marcus's steps, we know that .

Expand:

So, . To find , just take the square root:

So, Marcus travels exactly  back to the start.

### Example Question #54 : 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?

Explanation:

This is solving for the hypotenuse of a triangle. Using the Pythagorean Theorem, which says that

### Example Question #101 : 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?

70

100

200

50

25

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  and , respectively, making the hypotenuse equal to .

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

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

Susie will walk 100 meters to reach her house.

### Example Question #101 : Plane Geometry

The lengths of the sides of a triangle are consecutive odd numbers and the triangle's perimeter is 57 centimeters. What is the length, in centimeters, of its longest side?

23

21

19

25

17

21

Explanation:

First, define the sides of the triangle. Because the side lengths are consecutive odd numbers, if we define the shortest side will be as , the next side will be defined as , and the longest side will be defined as . We can then find the perimeter of a triangle using the following formula:

Substitute in the known values and variables.

Subtract 6 from both sides of the equation.

Divide both sides of the equation by 3.

Solve.

This is not the answer; we need to find the length of the longest side, or

Substitute in the calculated value for  and solve.

The longest side of the triangle is 21 centimeters long.

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