Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.9 AIME, harder Prove it

[ Area of a Trapezoid ]

Prove that if in trapezoid $A B C D$ the midpoint $M$ of one lateral side $A B$ is connected to the ends of the other lateral side $C D$ , then the area of the resulting triangle $C M D$ will be half the area of the trapezoid.

Solution

Prove that the sum of the areas of triangles $M B C$ and $M A D$ is equal to half the area of the given trapezoid.

## Solution

Let the height of the given trapezoid be $h$ . Then the heights of triangles $M B C$ and $M A D$ , drawn from vertex $M$ , are $\frac{h}{2}$ . The sum of the areas of these triangles is

$\frac{1}{2} \cdot \frac{1}{2} h \cdot B C + \frac{1}{2} \cdot \frac{1}{2} \cdot h \cdot A D = \frac{1}{4}(B C + A D) h$

which is half the area of the trapezoid. Therefore, the area of triangle $C M D$ is also half the area of the trapezoid.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.