A triangle has side lengths of x, x + 4, and 14 units, where x is an integer. Which of the… — GRE General Test

A triangle has side lengths of $x$ , $x + 4$ , and $14$ units, where $x$ is an integer. Which of the following could be the perimeter of the triangle? Select all that apply.

A
$26$
B
$28$
$30$ Answer
$34$ Answer
$40$ Answer

Answer

The possible perimeters of the triangle are 30, 34, and 40.

The Triangle Inequality Theorem dictates that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the remaining side. For side lengths

x

x + 4

, and

14

, setting

x + (x + 4) > 14

yields

2x > 10

, so

x > 5

. Since

x

is specified as an integer,

x

can be any integer greater than or equal to 6. Substituting valid values of

x

into the perimeter formula

P = 2x + 18

gives possible perimeters of 30 (when

x = 6

), 34 (when

x = 8

), and 40 (when

x = 11

). Thus, the values 30, 34, and 40 are all valid perimeters.

Step-by-Step Solution

Apply the Triangle Inequality Theorem

x > 5

According to the Triangle Inequality Theorem, the sum of any two side lengths must be strictly greater than the third side length. Therefore,

x + (x + 4) > 14

, which simplifies to

2x + 4 > 14 \implies 2x > 10 \implies x > 5

. The other two inequality conditions (

(x + 4) + 14 > x

and

x + 14 > x + 4

) are satisfied for all positive values of

x

Express the perimeter in terms of

x

P = 2x + 18

The perimeter

P

is the sum of the three side lengths:

P = x + (x + 4) + 14 = 2x + 18

Evaluate the given choices against the valid bounds of

x

Valid integer values of

x \ge 6

correspond to perimeters

P \ge 30

that are even numbers.

Since

x

is an integer and

x > 5

, the minimum integer value for

x

is 6, which yields a minimum perimeter of

P = 2(6) + 18 = 30

. Testing each option:
- For 26:

2x + 18 = 26 \implies x = 4

(Invalid,

x \le 5

)
- For 28:

2x + 18 = 28 \implies x = 5

(Invalid,

x \le 5

)
- For 30:

2x + 18 = 30 \implies x = 6

(Valid)
- For 34:

2x + 18 = 34 \implies x = 8

(Valid)
- For 40:

2x + 18 = 40 \implies x = 11

(Valid)

Key Concept

Triangle Inequality Theorem and Perimeter Calculation

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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