### All ACT Math Resources

## Example Questions

### Example Question #21 : Right Triangles

Given a right triangle with a leg length of and a hypotenuse length of , what is the height of the triangle?

**Possible Answers:**

**Correct answer:**

1. Use the pythagorean theorem with and :

2. Solve for :

(Notice that this right triangle is also a triangle.)

### Example Question #21 : Right Triangles

A ladder is leaning against a wall. If the bottom of the ladder touches the ground from the base of the wall, approximately (to the nearest whole number) how far is the top of the ladder from the base of the wall?

**Possible Answers:**

**Correct answer:**

To answer this, we first need to understand that we are trying to find the leg of a right triangle. The ladder, when leaned against the wall, forms a right triangle where the ladder is the hypotenuse, the wall is one leg, and the ground between the ladder and the wall is the other leg.

To find how high up the ladder touches the wall, we use the Pythagorean Theorem, which is:

is the length of the hypotenuse (the ladder itself), is the distance of the ladder base from the wall, and is how high the ladder touches on the wall.

We then rearrange to solve for one of the legs by subtracting from both sides.

We can now plug in our values of 10 for the hypotenuse and one of our legs (in this case, )

We can then take the square root of this equation to get an answer for

The question asked us to approximate, so we must round to the nearest whole number. To do this, we round a number up one place if the last digit is a 5, 6, 7, 8, or 9, and we round it down if the last digit is a 1, 2, 3, or 4. Therefore:

Therefore, our answer is .

### Example Question #81 : Plane Geometry

Right triangle has sides that are integers, and an area of . Which of the following could not be the height of ?

**Possible Answers:**

**Correct answer:**

The formula for the area of a triangle is , where is the length and is the height of the triangle. In this case, we know that , which means . Since we are told that the sides are integers, this means that must consist of two factors of . Since is the only number choice not a factor of , it is the only answer that could not meet the requirements.

### Example Question #82 : Plane Geometry

Find the height of a right triangle given a base of and hypotenuse of .

**Possible Answers:**

**Correct answer:**

To solve, simply realize this is one of our special right triangles, . Thus, our answer is . If you forgot this, you can use the Pythagorean Theorem.

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