5) In a right triangle, the projections of the legs on the hypotenuse measure $3 \mathrm{~m}$ and 12 $\mathrm{m}$ respectively. What is the area of the triangle?
(A) $45 \mathrm{~m}^{2}$,
(B) $60 \mathrm{~m}^{2}$
(C) $72 \mathrm{~m}^{2}$
(D) $84 \mathrm{~m}^{2}$
(E) $90 \mathrm{~m}^{2}$.
5. The answer is $(\mathbf{A})$.

The hypotenuse $A B$ has a length equal to the sum of the lengths of the projections of the legs, i.e., $3 \mathrm{~m}+12 \mathrm{~m}=15 \mathrm{~m}$. The length of the altitude $C H$ can be calculated using the second theorem of Euclid: $\overline{C H}^{2}=\overline{A H} \cdot \overline{H B}=36 \mathrm{~m}^{2}$.

Consequently, $\overline{C H}=6 \mathrm{~m}$ and the area is $\frac{\overline{A B} \cdot \overline{C H}}{2}=$ $45 \mathrm{~m}^{2}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_4701f996ee50400fb9c5g-2.jpg?height=291&width=526&top_left_y=757&top_left_x=1416)