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1977 AMC 12 Question 26

AMC12 · 1977 AMC 12

Question 26

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
  1. $A=\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)$ if
    and only if $MNPQ$ is convex
  2. $A=\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)$ if and
    only if $MNPQ$ is a rectangle
  3. $A\le\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)$ if
    and only if $MNPQ$ is a rectangle
  4. $A\le\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)$ if
    and only if $MNPQ$ is a parallelogram
  5. $A\ge\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)$ if
    and only if $MNPQ$ is a parallelogram