1952 AMC 12 Problem 46
Attempt Problem 46 of the 1952 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 1952 AMC 12 solutions, or check the answer key.
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46.
The base of a new rectangle equals the sum of the diagonal and the greater side of a given rectangle, while the altitude of the new rectangle equals the difference of the diagonal and the greater side of the given rectangle. The area of the new rectangle is:
Greater than the area of the given rectangle
Equal to the area of the given rectangle
Equal to the area of a square with its side
equal to the smaller side of the given rectangle
Equal to the area of a square with its side equal to the
greater side of the given rectangle
Equal to the area of a rectangle whose dimensions are the
diagonal and shorter side of the given rectangle
Answer: C
Small hint:
Let be the diagonal and the greater and smaller sides
Big hint:
The new area is
Solution:
Let the greater and smaller side lengths be and and let the diagonal be The new rectangle’s area is The Pythagorean theorem gives so the new area is This is the area of a square whose side equals the smaller side of the original rectangle.
Thus, the correct answer is C.