### All PSAT Math Resources

## Example Questions

### Example Question #1 : Right Triangles

Three points in the xy-coordinate system form a triangle.

The points are .

What is the perimeter of the triangle?

**Possible Answers:**

**Correct answer:**

Drawing points gives sides of a right triangle of 4, 5, and an unknown hypotenuse.

Using the pythagorean theorem we find that the hypotenuse is .

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

Based on the information given above, what is the perimeter of triangle ABC?

**Possible Answers:**

**Correct answer:**

Consult the diagram above while reading the solution. Because of what we know about supplementary angles, we can fill in the inner values of the triangle. Angles A and B can be found by the following reductions:

A + 120 = 180; A = 60

B + 150 = 180; B = 30

Since we know A + B + C = 180 and have the values of A and B, we know:

60 + 30 + C = 180; C = 90

This gives us a 30:60:90 triangle. Now, since 17.5 is across from the 30° angle, we know that the other two sides will have to be √3 and 2 times 17.5; therefore, our perimeter will be as follows:

### 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:**

85

75

90

80

95

**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 #2 : Right Triangles

and is a right angle.

Which angle or angles *must* be complementary to ?

I)

II)

III)

IV)

V)

**Possible Answers:**

II and V only

IV only

I only

I and III only

II 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 #11 : 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 #131 : Plane Geometry

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 #1 : How To Find If Right Triangles Are Similar

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,

### Example Question #1 : Right Triangles

A right triangle has one side equal to 5 and its hypotenuse equal to 14. Its third side is equal to:

**Possible Answers:**

171

14.87

9

12

13.07

**Correct answer:**

13.07

The Pythagorean Theorem gives us *a*^{2} + *b*^{2} = *c*^{2} for a right triangle, where *c* is the hypotenuse and *a* and *b* are the smaller sides. Here *a* is equal to 5 and *c* is equal to 14, so *b*^{2} = 14^{2} – 5^{2} = 171. Therefore *b* is equal to the square root of 171 or approximately 13.07.

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

Which of the following could NOT be the lengths of the sides of a right triangle?

**Possible Answers:**

5, 7, 10

8, 15, 17

14, 48, 50

5, 12, 13

12, 16, 20

**Correct answer:**

5, 7, 10

We use the Pythagorean Theorem and we calculate that 25 + 49 is not equal to 100.

All of the other answer choices observe the theorem *a*^{2} + *b*^{2} = *c*^{2}

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

Which set of sides could make a right triangle?

**Possible Answers:**

6, 7, 8

9, 12, 15

4, 6, 9

10, 12, 16

**Correct answer:**

9, 12, 15

By virtue of the Pythagorean Theorem, in a right triangle the sum of the squares of the smaller two sides equals the square of the largest side. Only 9, 12, and 15 fit this rule.

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