1986 AMC 12 Problem 26
Attempt Problem 26 of the 1986 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 1986 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
26.
It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the and axes and so that the medians to the midpoints of the legs lie on the lines and The number of different constants for which such a triangle exists is
more than
Answer: C
Small Hint:
Place an axis-aligned right triangle at convenient coordinates and compute the slopes of the two medians to its legs
Big Hint:
The two slopes differ by a factor of remember that either given line could be the steeper one
Solution:
Place the right-angle vertex at and the other vertices at and The medians to the legs have slopes and whose ratio is Therefore, if one median has slope the other can have slope or Both occur: choose a triangle with the required pair of slopes and translate its centroid to the intersection of the two specified lines. Thus there are two possible values of
Therefore the correct answer is C.
Problem 26 in Other Years
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