Problem 277

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Problem 277: (2007 AMC 12b, 20)

The parallelogram bounded by the lines $y=ax+c$, $y=ax+d$, $y=bx+c$, and $y=bx+d$ has area $18$. The parallelogram bounded by the lines $y=ax+c$, $y=ax-d$, $y=bx+c$, and $y=bx-d$ has area $72$. Given that $a$, $b$, $c$, and $d$ are positive integers, what is the smallest possible value of $a+b+c+d$? $\mathrm {(A)} 13\qquad$ $ \mathrm {(B)} 14\qquad$ $ \mathrm {(C)} 15\qquad$ $ \mathrm {(D)} 16\qquad$ $ \mathrm {(E)} 17$

Solution

Exercise 277: Solution

The parallelogram bounded by $y=ax+c$, $y=ax+d$, $y=bx+c$, and $y=bx+d$ has area $18$. The parallelogram bounded by $y=ax+c$, $y=ax-d$, $y=bx+c$, and $y=bx-d$ has area $50$. If $a,b,c,d$ are positive integers, what is the smallest possible value of $c+d$?

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