### All ACT Math Resources

## Example Questions

### Example Question #1 : Acute / Obtuse Isosceles Triangles

What is the area of an isosceles triangle with a vertex of degrees and two sides equal to ?

**Possible Answers:**

**Correct answer:**

Based on the description of your triangle, you can draw the following figure:

You can do this because you know:

- The two equivalent sides are given.
- Since a triangle is degrees, you have only or degrees left for the two angles of equal size. Therefore, those two angles must be degrees and degrees.

Now, based on the properties of an isosceles triangle, you can draw the following as well:

Based on your standard reference triangle, you know:

Therefore, is .

This means that is and the total base of the triangle is .

Now, the area of the triangle is:

or

### Example Question #1 : Acute / Obtuse Isosceles Triangles

An isosceles triangle has a height of and a base of . What is its area?

**Possible Answers:**

**Correct answer:**

Use the formula for area of a triangle:

### Example Question #1 : Acute / Obtuse Isosceles Triangles

An isosceles triangle has a base length of and a height that is twice its base length. What is the area of this triangle?

**Possible Answers:**

**Correct answer:**

1. Find the height of the triangle:

2. Use the formula for area of a triangle:

### Example Question #4 : Acute / Obtuse Isosceles Triangles

The height of an isosceles triangle, dropped from the vertex to its base, is one fourth the length of the base. If the area of this triangle is , what is its perimeter?

**Possible Answers:**

**Correct answer:**

Based on the description of this question, you can draw your triangle as such. We can do this thanks to the nature of an isosceles triangle:

Now, you know that the area of a triangle is defined as:

So, for our data, we can say:

Solving for , we get:

Thus, .

Now, for our little triangle on the right, we can draw:

Using the Pythagorean Theorem, we know that the other side is:

This can be simplified to:

Now, we know that this side is the "equal" side of the isosceles triangle. Therefore, we can know that the total perimeter is:

### Example Question #5 : Acute / Obtuse Isosceles Triangles

The base of an isosceles triangle is five times the length of its correlative height. If the area of this triangle is , what is its perimeter?

**Possible Answers:**

**Correct answer:**

Based on the description of this question, you can draw your triangle as such. We can do this thanks to the nature of an isosceles triangle:

Now, you know that the area of a triangle is defined as:

So, for our data, we can say:

Solving for , we get:

Thus, .

Now, for our little triangle on the right, we can draw:

Using the Pythagorean Theorem, we know that the other side is:

This can be simplified to:

Now, we know that this side is the "equal" side of the isosceles triangle. Therefore, we can know that the total perimeter is:

### Example Question #6 : Acute / Obtuse Isosceles Triangles

What is the area of an isosceles triangle with a vertex of degrees and two sides equal to units?

**Possible Answers:**

**Correct answer:**

Based on the description of your triangle, you can draw the following figure:

You can do this because you know:

- The two equivalent sides are given.
- Since a triangle is degrees, you have only or degrees left for the two angles of equal size. Therefore, those two angles must be degrees and degrees.

Now, based on the properties of an isosceles triangle, you can draw the following as well:

Based on your standard reference triangle, you know:

Therefore, is .

This means that is , and the total base of the triangle is .

Now, the area of the triangle is:

or

### Example Question #1 : Acute / Obtuse Isosceles Triangles

What is the perimeter of an isosceles triangle with a vertex of degrees and two sides equal to ?

**Possible Answers:**

**Correct answer:**

Based on the description of your triangle, you can draw the following figure:

You can do this because you know:

- The two equivalent sides are given.
- Since a triangle is degrees, you have only or degrees left for the two angles of equal size. Therefore, those two angles must be degrees and degrees.

Now, based on the properties of an isosceles triangle, you can draw the following as well:

Based on your standard reference triangle, you know:

Therefore, is .

This means that is and the total base of the triangle is .

Therefore, the perimeter of the triangle is:

### Example Question #8 : Acute / Obtuse Isosceles Triangles

Triangle A and Triangle B are similar isosceles triangles. Triangle A's sides measure , , and . Two of the angles in Triangle A each measure . Triangle B's sides measure , , and . What is the measure of the smallest angle in Triangle B?

**Possible Answers:**

**Correct answer:**

Because the interior angles of a triangle add up to , and two of Triangle A's interior angles measure , we must simply add the two given angles and subtract from to find the missing angle:

Therefore, the missing angle (and the smallest) from Triangle A measures . If the two triangles are similar, their interior angles must be congruent, meaning that the smallest angle is Triangle B is also .

The side measurements presented in the question are not needed to find the answer!

### Example Question #9 : Acute / Obtuse Isosceles Triangles

Triangle A and Triangle B are similar isosceles triangles. Triangle A has a base of and a height of . Triangle B has a base of . What is the length of Triangle B's two congruent sides?

**Possible Answers:**

**Correct answer:**

We must first find the length of the congruent sides in Triangle A. We do this by setting up a right triangle with the base and the height, and using the Pythagorean Theorem to solve for the missing side (). Because the height line cuts the base in half, however, we must use for the length of the base's side in the equation instead of . This is illustrated in the figure below:

Using the base of and the height of , we use the Pythagorean Theorem to solve for :

Therefore, the two congruent sides of Triangle A measure ; however, the question asks for the two congruent sides of Triangle B. In similar triangles, the ratio of the corresponding sides must be equal. We know that the base of Triangle A is and the base of Triangle B is . We then set up a cross-multiplication using the ratio of the two bases and the ratio of to the side we're trying to find (), as follows:

Therefore, the length of the congruent sides of Triangle B is .

### Example Question #221 : Plane Geometry

Isosceles triangles and share common side . is an obtuse triangle with sides . is also an obtuse isosceles triangle, where . What is the measure of ?

**Possible Answers:**

**Correct answer:**

In order to prove triangle congruence, the triangles must have SAS, SSS, AAS, or ASA congruence. Here, we have one common side (S), and no other demonstrated congruence. Hence, we cannot guarantee that side is not one of the two congruent sides of , so we cannot state congruence with .

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