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Difference between revisions of "1977 AHSME Problems/Problem 26"

(Created page with "==Problem== Let <math>a,b,c</math>, and <math>d</math> be the lengths of sides <math>MN,NP,PQ</math>, and <math>QM</math>, respectively, of quadrilateral <math>MNPQ</math>. If...")
 
(No difference)

Latest revision as of 14:10, 19 June 2021

Problem

Let $a,b,c$, and $d$ be the lengths of sides $MN,NP,PQ$, and $QM$, respectively, of quadrilateral $MNPQ$. If $A$ is the area of $MNPQ$, then

$\textbf{(A) }A=\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is convex}\\ \textbf{(B) }A=\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a rectangle}\\ \textbf{(C) }A\le\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a rectangle}\\ \textbf{(D) }A\le\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a parallelogram}\\ \textbf{(E) }A\ge\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a parallelogram}$

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