### All PSAT Math Resources

## Example Questions

### Example Question #561 : Geometry

A triangle has sides of length 5, 7, and x. Which of the following can NOT be a value of x?

**Possible Answers:**

5

11

3

13

7

**Correct answer:**

13

The sum of the lengths of any two sides of a triangle must exceed the length of the third side; therefore, 5+7 > x, which cannot happen if x = 13.

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

The lengths of two sides of a triangle are 9 and 7. Which of the following could be the length of the third side?

**Possible Answers:**

16

2

17

12

1

**Correct answer:**

12

Let us call the third side x. According to the Triangle Inequality Theorem, the sum of any two sides of a triangle must be larger than the other two sides. Thus, all of the following must be true:

x + 7 > 9

x + 9 > 7

7 + 9 > x

We can solve these three inequalities to determine the possible values of x.

x + 7 > 9

Subtract 7 from both sides.

x > 2

Now, we can look at x + 9 > 7. Subtracting 9 from both sides, we obtain

x > –2

Finally, 7 + 9 > x, which means that 16 > x.

Therefore, x must be greater than 2, greater than –2, but also less than 16. The only number that satisfies all of these requirements is 12.

The answer is 12.

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

The lengths of a triangle are 8, 12, and x. Which of the following inequalities shows all of the possible values of x?

**Possible Answers:**

8 < x < 12

4 < x <12

4 ≤ x ≤12

4 ≤ x ≤ 20

4 < x < 20

**Correct answer:**

4 < x < 20

According to the Triangle Inequality Theorem, the sum of any two sides of a triangle must be greater (not greater than or equal) than the remaining side. Thus, the following inequalities must all be true:

x + 8 > 12

x + 12 > 8

8 + 12 > x

Let's solve each inequality.

x + 8 > 12

Subtract 8 from both sides.

x > 4

Next, let's look at the inequality x + 12 > 8

x + 12 > 8

Subtract 12 from both sides.

x > –4

Lastly, 8 + 12 > x, which means that x < 20.

This means that x must be less than twenty, but greater than 4 and greater than –4. Since any number greater than 4 is also greater than –4, we can exclude the inequality x > –4.

To summarize, x must be greater than 4 and less than 20. We can write this as 4 < x < 20.

The answer is 4 < x < 20.

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

If 2 sides of the triangle are have lengths equal to 8 and 14, what is one possible length of the third side?

**Possible Answers:**

4

Not enough information

22

20

6

**Correct answer:**

20

The sum of the lengths of 2 sides of a triangle must be greater than—but not equal to—the length of the third side. Further, the third side must be longer than the difference between the greater and the lesser of the other two sides; therefore, 20 is the only possible answer.

### Example Question #7 : How To Find The Length Of The Side Of An Acute / Obtuse Triangle

In the length of AB is 15 and the length of side AC is 5. What is the least possible integer length of side BC?

**Possible Answers:**

9

11

10

13

17

**Correct answer:**

11

Rule - the length of one side of a triangle must be greater than the differnce and less than the sum of the lengths of the other two sides.

Given lengths of two of the sides of the are 15 and 5. The length of the third side must be greater than 15-5 or 10 and less than 15+5 or 20.

The question asks what is the least possible integer length of BC, which would be 11

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

Two sides of a triangle have lengths 4 and 7. Which of the following represents the set of all possible lengths of the third side, x?

**Possible Answers:**

3 < x < 7

4 < x < 7

2 < x < 12

3 < x < 11

4 < x < 11

**Correct answer:**

3 < x < 11

The set of possible lengths is: 7-4 < x < 7+4, or 3 < X < 11.

### Example Question #561 : Geometry

If two sides of a triangle have lengths 8 and 10, what could the length of the third side NOT be?

**Possible Answers:**

19

14

10

8

5

**Correct answer:**

19

According to the Triangle Inequality Theorem, the sums of the lengths of any two sides of a triangle must be greater than the length of the third side. Since 10 + 8 is 18, the only length out of the answer choices that is not possible is 19.

### Example Question #561 : Geometry

Which of the following describes a triangle with sides of length 14 inches, 18 inches, and 2 feet?

**Possible Answers:**

The triangle cannot exist.

The triangle is scalene and acute.

The triangle is scalene and obtuse.

The triangle is isosceles and acute.

The triangle is isosceles and obtuse.

**Correct answer:**

The triangle is scalene and obtuse.

2 feet is equal to 24 inches,so the sides of the triangle measure 14, 18, and 24.

First, we test to make sure the triangle exists. By the Triangle Inequality, for this to exist, the sum of the two smaller sidelengths must exceed the third, which we can see is the case:

The three sidelengths are all unequal, so the triangle is scalene.

Finally, we compare the sum of the squares of the smaller sidelengths to the square of the third:

Since

,

the triangle is obtuse.

The correct response is that the triangle is scalene and obtuse.

### Example Question #561 : Geometry

All of the following could be the possible side lengths of a triangle EXCEPT:

**Possible Answers:**

**Correct answer:**

The length of the third side of a triangle must always be between (but not equal to) the *sum* and the *difference* of the other two sides.

For instance, take the example of 2, 6, and 7.

and . Therefore, the third side length must be greater than 4 and less than 8. Because 7 is greater than 4 and less than 8, it is possible for these to be the side lengths of a triangle.

The 5, 7, 12 answer choice is the only option for which this is not the case.

and . Therefore, the third side length must be between 2 and 12. Because it is equal to the sum, not less than the sum, it is **not** possible that these could be the side lengths of a triangle.

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

and are similar triangles. The perimeter of Triangle A is 45” and the length of two of its sides are 15” and 10”. If the perimeter of Triangle B is 135” and what are lengths of two of its sides?

**Possible Answers:**

**Correct answer:**

The perimeter is equal to the sum of the three sides. In similar triangles, each side is in proportion to its correlating side. The perimeters are also in equal proportion.

Perimeter A = 45” and perimeter B = 135”

The proportion of Perimeter A to Perimeter B is .

This applies to the sides of the triangle. Therefore to get the any side of Triangle B, just multiply the correlating side by 3.

15” x 3 = 45”

10” x 3 = 30“