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Example Questions
Example Question #331 : Geometry
Given equilateral triangles and , construct the altitude from to on , and the altitude from to on .
Which, if either, of and is longer?
Statement 1:
Statement 2:
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Let and be the common side lengths of and . The length of an altitude of a triangle is solely a function of its side length, so it follows that the triangle with the greater side length is the one whose altitude is the longer. Therefore, the question is equivalent to which, if either, of or is the greater.
Assume Statement 1 alone. This statement can be rewritten as
It follows that has the greater side length, and, consequently, that its altitude is longer than .
Assume Statement 2 alone. divides the triangle into two congruent triangles, so is the midpoint of ; therefore, . Statement 2 can be rewritten as
This statement is inconclusive. Suppose —that is, each side of is of length 1. Then , , and all make that inequality true; without further information, it is therefore unclear whether , the side length of , is less than, equal to, or greater than , the side length of . Consequently, it is not clear which triangle has the longer altitude.