In triangle XYZ, the length of side XY is x + 3, the length of side YZ is 2x - 1, and the … — GRE General Test

In triangle $XYZ$ , the length of side $XY$ is $x + 3$ , the length of side $YZ$ is $2x - 1$ , and the length of side $XZ$ is $12$ , where $x$ is an integer. Which of the following could be the perimeter of triangle $XYZ$ ? Select all such perimeters.

26Answer
B
23
44Answer
D
62
E
31

Answer

The perimeters 26 and 44 are the valid values among the choices provided.

The perimeter formula

P = 3x + 14

must yield an integer value corresponding to an integer

x

in the range

4 \le x \le 15

. The perimeters equal to 26 (for

x = 4

) and 44 (for

x = 10

) fall within this valid range and satisfy the triangle inequality.

Step-by-Step Solution

Express the perimeter in terms of

x

\text{Perimeter } P = (x + 3) + (2x - 1) + 12 = 3x + 14

The perimeter of a triangle is the sum of its three side lengths.

Apply the Triangle Inequality Theorem to determine valid bounds for

x

Condition 1:

(x + 3) + (2x - 1) > 12 \implies 3x + 2 > 12 \implies 3x > 10 \implies x > 3.33

.
Condition 2:

(x + 3) + 12 > 2x - 1 \implies x + 15 > 2x - 1 \implies x < 16

.
Condition 3:

(2x - 1) + 12 > x + 3 \implies 2x + 11 > x + 3 \implies x > -8

(naturally satisfied for positive

x

).
Thus,

4 \le x \le 15

for integer

x

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

Evaluate the allowable perimeters for valid integer values of

x

The minimum valid perimeter corresponds to

x = 4

, giving

P = 3(4) + 14 = 26

.
The maximum valid perimeter corresponds to

x = 15

, giving

P = 3(15) + 14 = 59

.
Checking options:
- For

26

3x + 14 = 26 \implies x = 4

(valid).
- For

44

3x + 14 = 44 \implies x = 10

(valid).

Substituting valid integer

x

values identifies which proposed perimeters satisfy all conditions.

Key Concept

Triangle Inequality Theorem & Algebraic Bounds on Side Lengths

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic