## Task 5 - V10715

Construct a triangle from: $a=7 \mathrm{~cm}, b=6 \mathrm{~cm}, h_{a}=5 \mathrm{~cm}$!

How large is $h_{b}$? (Measurement and calculation!)

How many different triangles can be constructed with the given pieces? (Perform the construction!)
## Construction:

1. Draw the segment $B C$ of length $a$.
2. At an arbitrary point $F$ on $B C$, erect a perpendicular. On this perpendicular, let $H$ be a point at a distance $h_{a}$ from $F$.
3. Through $H$, construct the parallel line $p$ to $B C$.
4. A circle around $C$ with radius $b$ intersects the parallel line $p$ at two points. These points are $A$ and $A_{2}$.
5. The non-congruent triangles $A B C$ and $A_{2} B C$ are then the solutions to the problem.

Measuring the two heights $h_{b 1}$ and $h_{b 2}$ yields approximately $6 \mathrm{~cm}$ each. The calculation gives

$$
A=\frac{1}{2} a \cdot h_{a}=\frac{1}{2} b \cdot h_{b} \quad \rightarrow \quad h_{b}=\frac{a \cdot h_{a}}{b}=\frac{35}{6} \mathrm{~cm}
$$

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0307.jpg?height=711&width=893&top_left_y=547&top_left_x=593)