Track / Stage 4 / 93 of 340 #833 of 2444

Problem 833

AMC 12 late, AIME early

Geometry Difficulty 4.5 Prove it Annual Harvard-MIT November Tournament · United States

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 $ABC$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official 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 $CM \perp AB$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.