### All PSAT Math Resources

## Example Questions

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

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

**Possible Answers:**

85

95

80

75

90

**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 : Right Triangles

and is a right angle.

Which angle or angles *must* be complementary to ?

I)

II)

III)

IV)

V)

**Possible Answers:**

II only

IV only

I only

I and III only

II and V only

**Correct answer:**

II and V only

is a right angle, and, since corresponding angles of similar triangles are congruent, so is . A right angle cannot be part of a complementary pair so both can be eliminated.

can be eliminated, since it is congruent to ; congruent angles are not necessarily complementary.

Since is right angle, is a right triangle, and and are its acute angles. That makes complementary to . Since is congruent to , it is also complementary to .

The correct response is II and V only.

### Example Question #2 : Right Triangles

Refer to the above figure. Given that , give the perimeter of .

**Possible Answers:**

**Correct answer:**

By the Pythagorean Theorem,

The similarity ratio of to is

,

which is subsequently the ratio of the perimeter of to that of .

The perimeter of is

,

so the perimeter of can be found using this ratio:

### Example Question #2 : Right Triangles

Note: Figure NOT drawn to scale.

Refer to the above figure. Given that , give the area of .

**Possible Answers:**

The correct answer is not among the other responses.

**Correct answer:**

By the Pythagorean Theorem,

The similarity ratio of to is

,

This can be used to find :

The area of is therefore

### Example Question #3 : Right Triangles

Note: Figures NOT drawn to scale.

Refer to the above figure. Given that , evaluate .

**Possible Answers:**

**Correct answer:**

By the Pythagorean Theorem, since is the hypotenuse of a right triangle with legs 6 and 8, its measure is

.

The similarity ratio of to is

.

Likewise,