## 247. Math Puzzle $12 / 85$

A historical garden house needs to have its roof restored. It is a gable roof. The side lengths are $6 \mathrm{~m}$ and $9 \mathrm{~m}$. The ridge is $5 \mathrm{~m}$ long, and the height of the roof above the attic is $4 \mathrm{~m}$. The master roofer would like to know the area to be covered and encouragingly nudges his apprentice. The apprentice scratches his head sheepishly behind his right ear. Can you help him?
![](https://cdn.mathpix.com/cropped/2024_06_06_69be9c4355ed0dc2557cg-80.jpg?height=1002&width=1354&top_left_y=1578&top_left_x=338)
Solve and divide into triangles and trapezoids.

Trapezoid area $=\frac{9+5}{2} \cdot a ; a^{2}=4+b^{2}$ (Pythagoras); $b=\frac{6}{2}=3 ; a=5$

2 trapezoids $=\frac{9+5}{2} \cdot 5 \cdot 2=70 \mathrm{~m}^{2}$

Triangle area $\stackrel{6}{=} \frac{6 \cdot c}{2} ; c^{2}=4^{2}+d^{2}$ (Pythagoras); $d=\frac{9-5}{2}=2 \mathrm{~m} ; c=4.47 \mathrm{~m}$

2 triangles $=\frac{6 \cdot 4.47}{2} \cdot 2=26.8 \mathrm{~m}^{2}$

Total area $=96.8 \mathrm{~m}^{2}$. The roof area is $96.8 \mathrm{~m}^{2}$.