The shaded portion represents what percent of the area?

Question

In the figure below, the shaded portion represents what percent of the area of the triangle ABC?

1. D, E, F are midpoints of the sides of triangle
2. Triangle ABC is a equilateral triangle

adibapat · Answer

How is the answer for this A?
Shouldn't the triangle be equilateral as well to establish a proportion. Otherwise everything remains relative and dependant on the length of each side
 Bunuel

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yashikaaggarwal · Answer

adibapat  its not necessary for the triangle to be of equal sides. When 3 mid points of any triangle are connected, it divide triangle into 4 equal part by congruency of triangle. so the shaded region is 1/2 area of the whole triangle. you can draw any triangle by measure to justify the theory- scalene, isosceles, or equilateral triangle.

nick1816 · Answer

Statement 1-
Area of DEBF = (EB) * FB* (sin B) = 1/2 *BC *1/2*BA* sin B = 1/2 *   = 1/2 * area of ABC

Sufficient

Statement 2- We have no idea about  \frac{EB}{BC}  and  \frac{FB}{BA}

Insufficient

GMATGuruNY · Answer

Statement 1: triangle BEF.png In the figure above, E is the midpoint of BC and F is the midpoint of AB. TRIANGLES ABC AND BEF: Triangles ABC and BEF share angle EFB. Shared angle EFB is formed by sides in the same ratio (BC:BE = 2:1, AB:BF = 2:1). Rule: Triangles with a shared angle (EBF) formed by corresponding sides in the same ratio (BC:BE = 2:1, AB:BF = 2:1) are SIMILAR. Thus, triangle ABC is similar to triangle BEF. RULE FOR TWO SIMILAR TRIANGLES: If each side in the larger triangle is k times the corresponding side in the smaller triangle, then the area of the larger triangle is k^2 times the area of the smaller triangle. Here, since each side of ABC is 2 times the corresponding side in BEF, the area of ABC is 4 times the area of BEF. Thus, BEF = \frac{1}{4}ABC . The same line of reasoning can be used to show that ADF = \frac{1}{4}ABC and that CDE = \frac{1}{4}ABC , with the result that DEF = \frac{1}{4}ABC . Thus: shaded region = DEF + BEF = \frac{1}{4}ABC + \frac{1}{4}ABC = \frac{1}{2}ABC = 50 % SUFFICIENT.

bumpbot · Answer

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The shaded portion represents what percent of the area?

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The shaded portion represents what percent of the area?

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