### All ACT Math Resources

## Example Questions

### Example Question #1 : Right Triangles

Given a right triangle with a leg length of 6 and a hypotenuse length of 10, find the length of the other leg, *x*.

**Possible Answers:**

8

64

4

16

**Correct answer:**

8

Using Pythagorean Theorem, we can solve for the length of leg *x*:

*x*^{2} + 6^{2} = 10^{2}

Now we solve for *x*:

*x*^{2} + 36 = 100

*x*^{2} = 100 – 36

*x*^{2} = 64

*x* = 8

Also note that this is proportionally a 3/4/5 right triangle, which is very common. Always look out for a side-to-hypoteneuse ratio of 3/5 or 4/5, or a side-to-side ratio of 3/4, in any right triangle, so that you may solve such triangles rapidly.

### Example Question #4 : How To Find The Length Of The Side Of A Right Triangle

In a right triangle a hypotenuse has a length of 8 and leg has a length of 7. What is the length of the third side to the nearest tenth?

**Possible Answers:**

**Correct answer:**3.9

Using the pythagorean theorem, 8^{2}=7^{2}+x^{2}. Solving for x yields the square root of 15, which is 3.9^{}

### Example Question #5 : How To Find The Length Of The Side Of A Right Triangle

Given a right triangle with a leg length of 2 and a hypotenuse length of √8, find the length of the other leg, *x*.

**Possible Answers:**

10

√8

4

6

2

**Correct answer:**

2

Using Pythagorean Theorem, we can solve for the length of leg *x*:

*x*^{2} + 2^{2} = (√8)^{2 }= 8

Now we solve for *x*:

*x*^{2} + 4 = 8

*x*^{2} = 8 – 4

*x*^{2} = 4

*x* = 2

### Example Question #6 : How To Find The Length Of The Side Of A Right Triangle

The legs of a right triangle are and . Rounded to the nearest whole number, what is the length of the hypotenuse?

**Possible Answers:**

**Correct answer:**

Use the Pythagorean Theorem. The sum of both legs squared equals the hypotenuse squared.

### Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Find the length of segment .

**Possible Answers:**

**Correct answer:**

The length of segment is

Note that triangles and are both special, 30-60-90 right triangles. Looking specifically at triangle , because we know that segment has a length of 4, we can determine that the length of segment is 2 using what we know about special right triangles. Then, looking at triangle now, we can use the same rules to determine that segment has a length of

which simplifies to .

### Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

A handicap ramp is long, and a person traveling the length of the ramp goes up vertically. What horizontal distance does the ramp cover?

**Possible Answers:**

**Correct answer:**

In this case, we are already given the length of the hypotenuse of the right triangle, but the Pythagorean formula still helps us. Plug and play, remembering that must always be the hypotenuse:

State the theorem.

** **Substitute your variables.

Simplify.

Thus, the ramp covers of horizontal distance.