A triangle has integer side lengths a, b, and c such that its perimeter is 18. Which of th… — GRE General Test

A triangle has integer side lengths $a$ , $b$ , and $c$ such that its perimeter is $18$ . Which of the following could be the area of the triangle? Select all such values.

$12$ Answer
$6\sqrt{6}$ Answer
$9\sqrt{3}$ Answer
D
$4\sqrt{15}$
E
$8\sqrt{5}$

Answer

The possible areas of the triangle are

12

6\sqrt{6}

, and

9\sqrt{3}

To form a valid triangle with integer sides and a perimeter of 18, each side length must be a positive integer strictly less than 9 by the triangle inequality theorem. Evaluating Heron's formula

A = \sqrt{s(s-a)(s-b)(s-c)}

with

s=9

for all valid side combinations

(a,b,c)

produces the set of areas

\{3\sqrt{7}, 6\sqrt{3}, 3\sqrt{15}, 12, 6\sqrt{5}, 6\sqrt{6}, 9\sqrt{3}\}

. Among the options given, the values 12 (from side lengths 8, 5, 5),

6\sqrt{6}

(from side lengths 7, 6, 5), and

9\sqrt{3}

(from side lengths 6, 6, 6) are correct.

Step-by-Step Solution

Determine the structural bounds on side lengths using the Triangle Inequality Theorem.

For any side length

x

x < \frac{\text{Perimeter}}{2} = 9

. Thus,

a, b, c \in \{1, 2, 3, 4, 5, 6, 7, 8\}

The sum of any two side lengths must be strictly greater than the third side length.

List all unique integer side length combinations

(a, b, c)

where

a \ge b \ge c

and

a + b + c = 18

The valid triples are

(8, 8, 2)

(8, 7, 3)

(8, 6, 4)

(8, 5, 5)

(7, 7, 4)

(7, 6, 5)

, and

(6, 6, 6)

Exhaustive enumeration of integer partitions of 18 satisfying

a \le 8

Calculate the semi-perimeter

s

of the triangle.

s = \frac{18}{2} = 9

Required parameter for Heron's formula

A = \sqrt{s(s-a)(s-b)(s-c)}

Compute the area for candidate triples using Heron's formula.

For

(8, 5, 5)

A = \sqrt{9(1)(4)(4)} = 12

.
For

(7, 6, 5)

A = \sqrt{9(2)(3)(4)} = 6\sqrt{6}

.
For

(6, 6, 6)

A = \sqrt{9(3)(3)(3)} = 9\sqrt{3}

Direct evaluation of geometric area for the valid triangles.

Key Concept

Triangle Inequality Theorem and Area calculation via Heron's Formula

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic