Official Guide Explanation:
Data Sufficiency #109




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Solution and Metadata

Question: 109
Page: 282
Difficulty: 7 (Very Difficult)
Category 1: Geometry > Triangles > Multiple figures
Category 2: Geometry > Triangles > other

Explanation: An important first step is to draw a diagram to represent the description. Regardless of the shape of the triangle you draw, XYC will represent one corner of ABC, and RSC will be an even smaller corner of the triangle inside XYC. The important thing to recognize is that these three triangles are "similar." All three have the same angle (the angle at point C) between two sides that have the same ratio to each other. Since XC is half of AC and YC is half of BC, the ratio of XC to YC is the same as the ratio of AC to BC. The same reasoning goes for RC to SC.

Statement (1) is sufficient. ABX is exactly half the area of ABC -- it has two of the same sides as ABC, and the shorter side (AX) is half the length of the length of the corresponding side in ABC. Thus, if the area of ABX is half the area of ABC, so the area of ABC is 64. Thus, (1/2)bh = 64, or bh = 128. In RSC, each of the sides is one - quarter the length of the corresponding side in ABC. (This is where the "similar" triangles come in -- all of the measurements, not just RS and SC, are one - quarter of the corresponding measurement in ABC.) So we're looking for:




We know bh, so we can find this value.

Statement (2) is insufficient. It provides one useful piece of information towards finding the area of ABC, but to find the area of a triangle, we need both base and height ("altitude"), not just one. Choice (A) is correct.

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