1950 AMC 12 Problem 48
Attempt Problem 48 of the 1950 AMC 12 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 1950 AMC 12 solutions, or check the answer key.
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48.
A point is selected at random inside an equilateral triangle. From this point perpendiculars are dropped to each side. The sum of these perpendiculars is:
Least when the point is the center of gravity of the
triangle
Greater than the altitude of the triangle
Equal to the altitude of the triangle
One-half the sum of the sides of the triangle
Greatest when the point is the center of gravity
Answer: C
Small hint:
Join the interior point to all three vertices
Big hint:
Add the areas of the three smaller triangles using the common side length as their bases
Solution:
Let the equilateral triangle have side length and altitude and let the three perpendicular distances be Splitting the triangle at the interior point gives Therefore independent of the selected point.
Thus, the correct answer is C.