Difference between revisions of "2017 AMC 8 Problems/Problem 16"
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<math>\textbf{(A) }\frac{3}{4}\qquad\textbf{(B) }\frac{3}{2}\qquad\textbf{(C) }2\qquad\textbf{(D) }\frac{12}{5}\qquad\textbf{(E) }\frac{5}{2}</math> | <math>\textbf{(A) }\frac{3}{4}\qquad\textbf{(B) }\frac{3}{2}\qquad\textbf{(C) }2\qquad\textbf{(D) }\frac{12}{5}\qquad\textbf{(E) }\frac{5}{2}</math> | ||
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+ | ==Solution== | ||
We know that the perimeters of the two small triangles are <math>3+CD+AD</math> and <math>4+BD+AD.</math> Setting both equal and using <math>BD+CD = 5,</math> we have <math>BD = 2</math> and <math>CD = 3.</math> Now, we simply have to find the area of <math>\triangle ABD.</math> We can use <math>AB</math> as the base and the altitude from <math>D</math>. Let's call the foot of the altitude <math>E.</math> We have <math>\triangle BDE</math> similar to <math>BAC.</math> | We know that the perimeters of the two small triangles are <math>3+CD+AD</math> and <math>4+BD+AD.</math> Setting both equal and using <math>BD+CD = 5,</math> we have <math>BD = 2</math> and <math>CD = 3.</math> Now, we simply have to find the area of <math>\triangle ABD.</math> We can use <math>AB</math> as the base and the altitude from <math>D</math>. Let's call the foot of the altitude <math>E.</math> We have <math>\triangle BDE</math> similar to <math>BAC.</math> | ||
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+ | ==See Also== | ||
+ | {{AMC8 box|year=2017|num-b=15|num-a=17}} | ||
+ | |||
+ | {{MAA Notice}} |
Revision as of 14:44, 22 November 2017
Problem 16
In the figure below, choose point on so that and have equal perimeters. What is the area of ?
Solution
We know that the perimeters of the two small triangles are and Setting both equal and using we have and Now, we simply have to find the area of We can use as the base and the altitude from . Let's call the foot of the altitude We have similar to
See Also
2017 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
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