### All ACT Math Resources

## Example Questions

### Example Question #11 : Acute / Obtuse Triangles

Two similar triangles have perimeteres in the ratio . The sides of the smaller triangle measure , , and respectively. What is the perimeter, in meters, of the larger triangle?

**Possible Answers:**

**Correct answer:**

Since the perimeter of the smaller triangle is , and since the larger triangle has a perimeter in the ratio, we can set up the following identity, where the perimeter of the larger triangle:

In cross multiplying this identity, we get . We can now solve for . Here, , so the perimeter of the larger triangle is .

### Example Question #1 : How To Find The Length Of The Hypotenuse Of An Acute / Obtuse Triangle

What is the value of in the triangle above? Round to the nearest hundredth.

**Possible Answers:**

Cannot be calculated

**Correct answer:**

Begin by filling in the missing angle for your triangle. Since a triangle has a total of degrees, you know that the missing angle is:

Draw out the figure:

Now, to solve this, you will need some trigonometry. Use the Law of Sines to calculate the value:

Solving for , you get:

Rounding, this is .

### Example Question #11 : Acute / Obtuse Triangles

The base of a triangle is and the area is . The height of the triangle is then decreased by . What is the final area of the triangle?

**Possible Answers:**

**Correct answer:**

The formula for the area of a triangle is

.

If the area is equal to 48 cm^{2} and the base is 8 cm, then the initial height is:

If 12 is decreased by 75% then

, and . The final height is 3 cm.

Therefore the final area is

.

### Example Question #1 : How To Find The Height Of An Acute / Obtuse Triangle

If the area of an isosceles triangle is and its base is , what is the height of the triangle?

**Possible Answers:**

**Correct answer:**

Use the formula for area of a triangle to solve for the height:

### Example Question #1 : How To Find The Height Of An Acute / Obtuse Triangle

Find the height of the isosceles triangle above if the length of and . If your answer is in a decimal form, round to the nearest tenths place.

**Possible Answers:**

**Correct answer:**

Because this is an isosceles triangle, . Also, we know that the base of the triangle, . Therefore, we create two triangles by bisecting the trinagle down the center. We are solving for the length of the long arm of the triangle. We know that the hypotenuse is 12 and the base is 4 (half of 8).

Thus we use the Pythagorean Theorem to find the length of the long arm:

### Example Question #3 : How To Find The Height Of An Acute / Obtuse Triangle

The triangle above has an area of units squared. If the length of the base is units, what is the height of the triangle?

**Possible Answers:**

**Correct answer:**

The area of a triangle is found using the formula

The height of any triangle is the length from it's highest point to the base, as pictured below:

We can find the height by rearranging the area formula:

### Example Question #81 : Triangles

The ratio of the side lengths of a triangle is 7:10:11. In a similar triangle, the middle side is 9 inches long. What is the length of the longest side of the second triangle?

**Possible Answers:**

7.7

10

9.9

9

12.1

**Correct answer:**

9.9

Side lengths of similar triangles can be expressed in proportions. Establish a proportion comparing the middle and long sides of your triangles.

10/11 = 9/x

Cross multiply and solve for x.

10x = 99

x = 9.9

### Example Question #3 : How To Find If Two Acute / Obtuse Triangles Are Similar

Two triangles are similar to each other. The bigger one has side lengths of 12, 3, and 14.

The smaller triangle's shortest side is 1 unit in length. What is the length of the smaller triangle's longest side?

**Possible Answers:**

**Correct answer:**

Because the triangles are similar, a ratio can be set up between the triangles' longest sides and shortest sides as such: 14/3 = x/1. Solving for x, we obtain that the shortest side of the triangle is 14/3 units long.

### Example Question #181 : Plane Geometry

The ratio of the side lengths of a triangle is . The longest side of a similar triangle is . What is the length of the smallest side of that similar triangle?

**Possible Answers:**

**Correct answer:**

Use proportions to solve for the missing side:

Cross multiply and solve:

### Example Question #182 : Plane Geometry

There are two similar triangles. One has side lengths of 14, 17, and 19. The smaller triangle's smallest side length is 2. What is the length of its longest side?

**Possible Answers:**

**Correct answer:**

Use proportions to solve for the missing side:

Cross multiply and solve:

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