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Area of Triangle

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by Mission2012 » Thu Aug 15, 2013 6:43 pm
What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius 1 and the other two vertices on the circle?

(A) 3√4

(B) 12

(C) π4

(D) 1

(E)√2
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by ganeshrkamath » Thu Aug 15, 2013 8:03 pm
Mission2012 wrote:What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius 1 and the other two vertices on the circle?

(A) 3√4

(B) 12

(C) π4

(D) 1

(E)√2

Area = 1/2 * base * height
Maximize the product:
base = 1
height = 1
Maximum area = 1/2 * 1 * 1
= 1/2
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by GMATGuruNY » Thu Aug 15, 2013 8:18 pm
Mission2012 wrote:What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius 1 and the other two vertices on the circle?

(A) √3/4

(B) 12

(C) 4Ï€

(D) 1

(E)√2
Image

The drawings above show 3 different versions of the triangle.

Leftmost drawing: b=1, h=1.
Middle drawing: b=1, h<1.
Rightmost drawing: b=1, h<1.

Notice that in each triangle b=1, but only in the leftmost triangle does h=1. In the other two triangles, h<1, resulting in a smaller area. The drawings above illustrate the following rule:

Given two sides of a triangle, the greatest area will be achieved when a right angle is placed between them (as in the leftmost triangle).

Thus, the greatest possible area = 1/2 * 1 * 1 = 1/2.

The correct answer is B.
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