### All PSAT Math Resources

## Example Questions

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

An isosceles triangle has an area of 12. If the ratio of the base to the height is 3:2, what is the length of the two equal sides?

**Possible Answers:**

4

5

6

4√3

3√3

**Correct answer:**

5

Area of a triangle is ½ x base x height. Since base:height = 3:2, base = 1.5 height. Area = 12 = ½ x 1.5 height x height or 24/1.5 = height^{2}. Height = 4. Base = 1.5 height = 6. Half the base and the height form the legs of a right triangle, with an equal leg of the isosceles triangle as the hypotenuse. This is a 3-4-5 right triangle.

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

Two sides of a triangle each have length 6. All of the following could be the length of the third side EXCEPT

**Possible Answers:**

**Correct answer:**12

This question is about the Triangle Inequality, which states that in a triangle with two sides A and B, the third side must be greater than the absolute value of the difference between A and B and smaller than the sum of A and B.

Applying the Triangle Inequality to this problem, we see that the third side must be greater than the absolute value of the difference between the other two sides, which is |6-6|=0, and smaller than the sum of the two other sides, which is 6+6=12. The only answer choice that does not satisfy this range of possible values is 12 since the third side must be LESS than 12.

### Example Question #1 : How To Find If Two Acute / Obtuse Triangles Are Similar

A triangle with two equal angles is called a(n) __________.

**Possible Answers:**

disjoint triangle

isosceles triangle

equilateral triangle

Pythagoras triangle

right triangle

**Correct answer:**

isosceles triangle

An isoceles triangle is a triangle that has at least two congruent sides (and therefore, at least two congruent angles as well).

### Example Question #1 : 45/45/90 Right Isosceles Triangles

An isosceles triangle has a base of 6 and a height of 4. What is the perimeter of the triangle?

**Possible Answers:**

None of these

**Correct answer:**

An isosceles triangle is basically two right triangles stuck together. The isosceles triangle has a base of 6, which means that from the midpoint of the base to one of the angles, the length is 3. Now, you have a right triangle with a base of 3 and a height of 4. The hypotenuse of this right triangle, which is one of the two congruent sides of the isosceles triangle, is 5 units long (according to the Pythagorean Theorem).

The total perimeter will be the length of the base (6) plus the length of the hypotenuse of each right triangle (5).

5 + 5 + 6 = 16

### Example Question #1 : Isosceles Triangles

What is the area of a square that has a diagonal whose endpoints in the coordinate plane are located at (-8, 6) and (2, -4)?

**Possible Answers:**

200√2

50√2

100

50

100√2

**Correct answer:**

100

### 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:

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