2011 AMC 8 Problem 16
Attempt Problem 16 of the 2011 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2011 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
16.
Let be the area of the triangle with sides of length and Let be the area of the triangle with sides of length and What is the relationship between and
Answer: C
Small hint:
The two right triangles use legs in opposite roles.
Big hint:
Drop an altitude in each isosceles triangle.
Solution:
Since these triangles are isosceles, the altitude shown from the apex of each triangle bisects its base and creates two congruent right triangles.
Using the Pythagorean theorem, we get that the altitude of the triangle with area equals Similarly, we get that the altitude of the triangle with area equals
With these altitudes, we can calculate the areas of the triangles. We get that Similarly,
Therefore,
Thus, C is the correct answer.
Problem 16 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
- 1993 AMC 8
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