2000 AMC 8 Problems/Problem 15
Contents
Problem
Triangles , , and are all equilateral. Points and are midpoints of and , respectively. If , what is the perimeter of figure ?
Solution 1
The large triangle has sides of length . The medium triangle has sides of length . The small triangle has sides of length . There are segment sizes, and all segments depicted are one of these lengths.
Starting at and going clockwise, the perimeter is:
, thus the answer is
Solution 2
The perimeter of is the perimeter of the three triangles, minus segments and , which are on the interior of the figure. Because each of these segments is on two triangles, each segment must be subtracted two times.
As in solution 1, the sides of the triangles are and , and the perimeters of the triangles are thus and .
The perimeter of the three triangles is . Subtracting the two segments and two times, the perimeter of is , and the answer is .
See Also
2000 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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