### All High School Math Resources

## Example Questions

### Example Question #81 : Triangles

You have two right triangles that are similar. The base of the first is 6 and the height is 9. If the base of the second triangle is 20, what is the height of the second triangle?

**Possible Answers:**

35

30

23

25

33

**Correct answer:**

30

Similar triangles are proportional.

Base_{1} / Height_{1} = Base_{2} / Height_{2}

6 / 9 = 20 / Height_{2}

Cross multiply and solve for Height_{2}

6 / 9 = 20 / Height_{2}

6 * Height_{2}= 20 * 9

Height_{2}= 30

### Example Question #1 : Right Triangles

In the figure above, line segments *DC* and *AB* are parallel. What is the perimeter of quadrilateral *ABCD*?

**Possible Answers:**

80

90

95

85

75

**Correct answer:**

85

Because *DC* and *AB* are parallel, this means that angles *CDB* and *ABD* are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as *CDB* and *ABD*) are congruent.

Now, we can show that triangles *ABD* and *BDC* are similar. Both *ABD* and *BDC* are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles *CDB* and *ABD* are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles *ABD* and *BDC* are similar triangles.

We can use the similarity between triangles *ABD* and *BDC* to find the lengths of *BC* and *CD*. The length of *BC* is proportional to the length of *AD*, and the length of *CD* is proportional to the length of *DB*, because these sides correspond.

We don’t know the length of *DB*, but we can find it using the Pythagorean Theorem. Let *a*, *b*, and *c* represent the lengths of *AD*, *AB*, and *BD* respectively. According to the Pythagorean Theorem:

*a*^{2} + *b*^{2 }= *c*^{2}

15^{2} + 20^{2 }= *c*^{2}

625 = *c*^{2}

*c* = 25

The length of *BD* is 25.

We now have what we need to find the perimeter of the quadrilateral.

Perimeter = sum of the lengths of *AB*, *BC*, *CD*, and *DA*.

Perimeter = 20 + 18.75 + 31.25 + 15 = 85

The answer is 85.

### Example Question #1 : How To Find If Right Triangles Are Similar

A right triangle is defined by the points (1, 1), (1, 5), and (4, 1). The triangle's sides are enlarged by a factor of 3 to form a new triangle. What is the area of the new triangle?

**Possible Answers:**

81 square units

54 square units

None of the answers are correct

36 square units

108 square units

**Correct answer:**

54 square units

The points define a 3-4-5 right triangle. Its area is A = 1/2bh = ½(3)(4) = 6. The scale factor (SF) of the new triangle is 3. The area of the new triangle is given by A_{new} = (SF)^{2} x (A_{old}) =

3^{2} x 6 = 9 x 6 = 54 square units (since the units are not given in the original problem).

NOTE: For a volume problem: V_{new} = (SF)^{3} x (V_{old}).