sandy wrote:
Which of the folllowing 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
Since the isosceles triangle has a perimeter of 18 and a side length of 8, we could have the following sides:
1) 8, 8, 2
or
2) 8, 5, 5
In option 1, the base is 2 and the legs (the sides that have equal length) are 8 each. The height of this triangle, h, satisfies the Pythagorean theorem in the form (b/2)^2 + h^2 = l^2 where b is the base and l is a leg of the isosceles triangle. Thus:
(2/2)^2 + h^2 = 8^2
1 + h^2 = 64
h^2 = 63
h = √63
Recall that the area of a triangle is (b x h)/2, so the area of the triangle is (2 x √63)/2 = √63. However, this is not one of the answer choices. Thus, we must consider option 2.
In option 2, the base is 8 and the legs are 5 each. So, we 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
So, the area of the triangle is (8 x 3)/2 = 24/2 = 12.
Answer: B