How to find the area of an equilateral triangle - ISEE Upper Level (grades 9-12) Quantitative Reasoning
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Question
is an equilateral triangle. Points 
are the midpoints of 
, respectively. 
is constructed.
Which is the greater quantity?
(a) The area of 
(b) Twice the area of 
Which is the greater quantity?
(a) The area of
(b) Twice the area of
Answer
If segments are constructed in which the endpoints form the midpoints of the sides of a triangle, then four triangles, congruent to each other and similar to the larger triangle, are formed. Therefore, one of these triangles - specifically, 
- would have one-fourth the area of 
. This means 
has more than twice the area of 
.
Note that the fact that the triangle is equilateral is irrelevant.
If segments are constructed in which the endpoints form the midpoints of the sides of a triangle, then four triangles, congruent to each other and similar to the larger triangle, are formed. Therefore, one of these triangles - specifically,
Note that the fact that the triangle is equilateral is irrelevant.
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