Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.5 AIME Prove it United States

Problem:

Let $ABC$ be a triangle, and let $M$ be the midpoint of side $AB$ . If $AB$ is $17$ units long and $CM$ is $8$ units long, find the maximum possible value of the area of $ABC$ .

Solution

Solution:

Answer: $68$

Let $h$ be the length of the altitude from $C$ to $AB$ . Observe that $K = \frac{1}{2} \cdot h \cdot AB \leq \frac{1}{2} \cdot CM \cdot AB = 68$ and that equality is achieved when $CM \perp AB$ .

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