Question: Which of the following could be the area of an isosceles triangle with perimeter 18 and one side of length 8?

(A) 6
(B) 12
(C) 14
(D) 16
(E) 18

Correct Answer: B

Solution and Explanation:
Approach Solution 1:

In an isosceles triangle, two of its sides are of the same length. We are aware that the triangle's perimeter is 18, 2x + y = 18, where x is the length of one of the sides and y is the length of the other.

We now understand that one of the sides is 8.

When either x or y equals 8, there are two options.
The three sides are 8,8,2 for x = 8.
The list of possible solutions does not include the area determined using Heron's formula.

The three sides are 5,5,8 if y = 8.
Because it will result in two right triangles with sides of 3, 4, and 12, the area is 2124321243.
The area must therefore be 12 (Option B)

Approach Solution 2:

The isosceles triangle can have any of the following sides given its 18-inch circumference and 8-inch side length:

1) 8, 8, 2
or
2) 8, 5, 5

Option 1 has a base of 2 and equal-length legs (sides) that are each 8. The isosceles triangle's height, h, satisfy the Pythagorean theorem with the formula (b/2)2 + h2 = l2, where b is the base and l is the leg. Thus:

(2/2)^2 + h^2 = 8^2
1 + h^2 = 64
h^2 = 63
h = √63

Remember that the area of a triangle is equal to (b x h)/2, so the triangle's area is equal to (2 x 63)/2, or 63. This, however, is not an option for an answer. We must therefore think about option 2.

Option 2 has a base of 8 and legs of 5 each. We thus have:

(b/2)^2 + h^2 = l^2
(8/2)^2 + h^2 = 5^2
16 + h^2 = 25
h^2 = 9
h = √9 = 3

Thus, the triangle's area is (8 x 3)/2 = 24/2 = 12.

“Which of the following could be the area of an isosceles triangle with" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, Candidates must have basic qualitative abilities. Quant evaluates a candidate's aptitude for both mathematics and logic. The problem-solving section of the GMAT Quantitative test consists of a question and a list of potential answers. The candidate must choose the right answer by applying maths to the question. The problem-solving section of the GMAT Quant topic is made up of very complicated maths problems that must be solved by using the right maths facts.

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