1965 AMC 12 Problem 25
Attempt Problem 25 of the 1965 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 1965 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
25.
Let be a quadrilateral with extended to so that Lines and are drawn to form angle For this angle to be a right angle it is necessary that quadrilateral have:
all angles equal
all sides equal
two pairs of equal sides
one pair of equal sides
one pair of equal angles
Answer: D
Small hint:
Because point is the midpoint of
Big hint:
In a right triangle, the midpoint of the hypotenuse is equidistant from all three vertices
Solution:
If then is the hypotenuse of right triangle Its midpoint is equidistant from Thus so quadrilateral necessarily has at least one pair of equal sides. No condition involving follows.
Therefore, the correct answer is D.
Problem 25 in other years
- 1950 AMC 12
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