2015 AMC 12A Problem 8

The ratio of the length to the width of a rectangle is

4 : 3.

If the rectangle has diagonal of length

d,

then the area may be expressed as

kd^2

for some constant

k.

What is

k?

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Solution

Let the sides of the rectangle be

4a

and

3a.

By the Pythagorean Theorem the diagonal is

5a = d,

a = \dfrac{d}{5}.

The area is

4a\cdot 3a = 12a^2

= 12\left(\dfrac{d}{5}\right)^2

= \dfrac{12}{25}d^2,

k = \dfrac{12}{25}.

Thus, the correct answer is C.