14. A teacher assigned the task of drawing a triangle $A B C$ at will on the notebook, drawing the three altitudes $A A^{\prime}, B B^{\prime}, C C^{\prime}$ and measuring their respective lengths. Five students found the measurements reported in the answers below. Which of them has definitely made a mistake?

(A) $\overline{A A^{\prime}}=4 \mathrm{~cm} \quad \overline{B B^{\prime}}=6 \mathrm{~cm} \quad \overline{C C^{\prime}}=8 \mathrm{~cm}$

(B) $\overline{A A^{\prime}}=3 \mathrm{~cm} \quad \overline{B B^{\prime}}=4 \mathrm{~cm} \quad \overline{C C^{\prime}}=6 \mathrm{~cm}$

(C) $\overline{A A^{\prime}}=3 \mathrm{~cm} \quad \overline{B B^{\prime}}=4 \mathrm{~cm} \quad \overline{C C^{\prime}}=8 \mathrm{~cm}$

(D) $\overline{A A^{\prime}}=4 \mathrm{~cm} \quad \overline{B B^{\prime}}=6 \mathrm{~cm} \quad \overline{C C^{\prime}}=9 \mathrm{~cm}$

(E) $\overline{A A^{\prime}}=3 \mathrm{~cm} \quad \overline{B B^{\prime}}=6 \mathrm{~cm} \quad \overline{C C^{\prime}}=8 \mathrm{~cm}$
Problem 14. The correct answer is (E).

Let $a^{\prime}, b^{\prime}, c^{\prime}$ be the measures of the three heights, then the measures $a, b, c$ of the corresponding three sides will be $a=\frac{2 \mathcal{A}}{a^{\prime}}, \quad b=\frac{2 \mathcal{A}}{b^{\prime}}, \quad c=\frac{2 \mathcal{A}}{c^{\prime}}, \quad$ where $\mathcal{A}$ denotes the area of the triangle. In the five cases considered, the triples of the sides should therefore be:

$\left(\frac{2 \mathcal{A}}{4}, \frac{2 \mathcal{A}}{6}, \frac{2 \mathcal{A}}{8}\right) \quad\left(\frac{2 \mathcal{A}}{3}, \frac{2 \mathcal{A}}{4}, \frac{2 \mathcal{A}}{6}\right) \quad\left(\frac{2 \mathcal{A}}{3}, \frac{2 \mathcal{A}}{4}, \frac{2 \mathcal{A}}{8}\right) \quad\left(\frac{2 \mathcal{A}}{4}, \frac{2 \mathcal{A}}{6}, \frac{2 \mathcal{A}}{9}\right) \quad\left(\frac{2 \mathcal{A}}{3}, \frac{2 \mathcal{A}}{6}, \frac{2 \mathcal{A}}{8}\right)$.

It is verified that: $\quad \frac{1}{4}\frac{1}{6}+\frac{1}{8}$.

The values $3,6,8$ cannot therefore be the measures of the three heights of a triangle, as it would violate the triangle inequality.