### All ACT Math Resources

## Example Questions

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

A right triangle has a total perimeter of 12, and the length of its hypotenuse is 5. What is the area of this triangle?

**Possible Answers:**

6

3

15

10

12

**Correct answer:**

6

The area of a triangle is denoted by the equation 1/2 b x h.

b stands for the length of the base, and h stands for the height.

Here we are told that the perimeter (total length of all three sides) is 12, and the hypotenuse (the side that is neither the height nor the base) is 5 units long.

So, 12-5 = 7 for the total perimeter of the base and height.

7 does not divide cleanly by two, but it does break down into 3 and 4,

and 1/2 (3x4) yields 6.

Another way to solve this would be if you recall your rules for right triangles, one of the very basic ones is the 3,4,5 triangle, which is exactly what we have here

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

The ratio for the side lengths of a right triangle is 3:4:5. If the perimeter is 48, what is the area of the triangle?

**Possible Answers:**

108

48

96

240

50

**Correct answer:**

96

We can model the side lengths of the triangle as 3x, 4x, and 5x. We know that perimeter is 3x+4x+5x=48, which implies that x=4. This tells us that the legs of the right triangle are 3x=12 and 4x=16, therefore the area is A=1/2 bh=(1/2)(12)(16)=96.

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

The length of one leg of an equilateral triangle is 6. What is the area of the triangle?

**Possible Answers:**

**Correct answer:**

The base is equal to 6.

The height of an quilateral triangle is equal to , where is the length of the base.

### Example Question #14 : How To Find The Area Of A Right Triangle

Find the area of the following right triangle to the nearest integer

Note: The triangle is not necessarily to scale

**Possible Answers:**

None of the other answers

**Correct answer:**

The equation used to find the area of a right triangle is: where A is the area, b is the base, and h is the height of the triangle. In this question, we are given the height, so we need to figure out the base in order to find the area. Since we know both the height and hypotenuse of the triangle, the quickest way to finding the base is using the pythagorean theorem, . a = the height, b = the base, and c = the hypotenuse.

Using the given information, we can write . Solving for b, we get or . Now that we have both the base and height, we can solve the original equation for the area of the triangle.

### Example Question #71 : Plane Geometry

In the above triangle, if and what is the area of the triangle?

**Possible Answers:**

**Correct answer:**

To find the area of a triangle use the formula:

, since the base is and the height is , plugging in yields:

### Example Question #71 : Geometry

A right triangle has integer sides with a ratio of , measured in . What is the smallest possible area of this triangle?

**Possible Answers:**

**Correct answer:**

The easiest way to find the smallest possible integer sides is to simply factor the ratio we are given. In this case, is already prime (since is a prime number), so the smallest possible sides which hold to this triangle are and . You may also recognize this number as a common Pythagorean triple.

The area of a triangle is expressed as , where is the length and is the height. Since our triangle is right, we know that two lines intersect at a angle and thus serve well as our length and height. We also know that the longest side is always the hypotenuse, so the other two sides must be and .

Applying our formula, we get:

Thus, the smallest possible area for our triangle is .

### Example Question #13 : Triangles

Right triangle has hypotenuse cm and . Find the area of the triangle, in cm^{2}, by using .

Round angles to four significant figures. Round side lengths to the nearest integer.

**Possible Answers:**

**Correct answer:**

To find the area of a right triangle, find the lengths of the two perpendicular legs (since this gives us our length and height for the area formula).

In this case, we know that one angle is , and SOHCAHTOA tells us that , so:

Substitute the angle measure and hypotenuse into the formula.

Isolate the variable.

Solve the left side (rounding to the nearest integer) using our Pythagorean formula:

---> Substitute known values.

** ---> **Simplify.

** **Square root both sides.

So with our two legs solved for, we now only need to apply the area formula for triangles to get our answer:

So, the area of our triangle is .

### Example Question #14 : Triangles

Find the area of a right triange whose height is 4 and base is 5.

**Possible Answers:**

**Correct answer:**

To solve, simply use the formula for the area of a triangle given height h and base B.

Substitute

and into the area formula.

Thus,