2008 AMC 10B Problems/Problem 7

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Problem

An equilateral triangle of side length $10$ is completely filled in by non-overlapping equilateral triangles of side length $1$ . How many small triangles are required?

$\mathrm{(A)}\ 10\qquad\mathrm{(B)}\ 25\qquad\mathrm{(C)}\ 100\qquad\mathrm{(D)}\ 250\qquad\mathrm{(E)}\ 1000$

Solution

The area of the large triangle is $\frac{10^2\sqrt3}{4}$ , while the area each small triangle is $\frac{1^2\sqrt3}{4}$ . Dividing these two quantities, we get 100, therefore $\boxed{100}$ small triangles can fit in the large one.

Another Solution:

The number of triangles is $1+3+\dots+19 = \boxed{100}$ .

2008 AMC 10B (Problems • Resources)
Preceded by Problem 6	Followed by Problem 8
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2008 AMC 10B Problems/Problem 7

From AoPSWiki

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