A triangle has side lengths of 8, 11, and x, where x is an integer. If the perimeter of th… — GRE General Test

A triangle has side lengths of $8$ , $11$ , and $x$ , where $x$ is an integer. If the perimeter of the triangle is a positive integer multiple of $5$ , which of the following could be the value of $x$ ? Select all such values.

A
$1$
$6$ Answer
$11$ Answer
$16$ Answer
E
$21$

Answer

The possible values for

x

are

6

11

, and

16

According to the Triangle Inequality Theorem, the third side

x

must be strictly greater than

11 - 8 = 3

and strictly less than

11 + 8 = 19

. The perimeter of the triangle is

8 + 11 + x = 19 + x

. For

19 + x

to be a positive multiple of

5

19 + x

can be

25

30

, or

35

within the allowed range for

x

, giving

x = 6

x = 11

, and

x = 16

Step-by-Step Solution

Apply the Triangle Inequality Theorem to determine the valid range for the unknown side

x

The length of

x

must satisfy

11 - 8 < x < 11 + 8

, which simplifies to

3 < x < 19

The sum of any two side lengths of a non-degenerate triangle must be strictly greater than the third side length.

Set up an expression for the perimeter

P

of the triangle.

P = 8 + 11 + x = 19 + x

Perimeter is the total sum of all three side lengths.

Determine which values of

x

within the range

3 < x < 19

make

P = 19 + x

a multiple of

5

x = 6

P = 25

(multiple of

5

). If

x = 11

P = 30

(multiple of

5

). If

x = 16

P = 35

(multiple of

5

Adding

19

6

11

, and

16

gives multiples of

5

within the strict inequality bounds.

Test boundary values outside the inequality bounds.

x = 1

gives

P = 20

, but

1 \le 3

(invalid).

x = 21

gives

P = 40

, but

21 \ge 19

(invalid).

Values outside

3 < x < 19

cannot form a valid triangle.

Key Concept

Triangle Inequality Theorem and Perimeter Constraints

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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