Question

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

Concepts:rectangle Pythagorean Theorem difference of squares

Difficulty rating: 1500

Small Hint:

Let $d$ be the diagonal and $L,W$ the greater and smaller sides

Big Hint:

The new area is $(d+L)(d-L)=d^2-L^2$

Solution:

Let the greater and smaller side lengths be $L$ and $W,$ and let the diagonal be $d.$ The new rectangle’s area is $(d+L)(d-L)=d^2-L^2.$ The Pythagorean theorem gives $d^2=L^2+W^2,$ so the new area is $W^2.$ This is the area of a square whose side equals the smaller side of the original rectangle.

Thus, the correct answer is C.

1952 AMC 12 Problem 46

Answer: C

Difficulty rating: 1500

Small Hint:

Big Hint:

Solution:

Problem 46 in Other Years