[Pythagorean Theorem (direct and inverse).]

Perpendiculars and two obliques are drawn from one point to a given straight line.

Find the length of the perpendicular, if the obliques are 41 and 50, and their projections on the given straight line are in the ratio $3: 10$
Express the desired perpendicular using the Pythagorean theorem from two right triangles and solve the resulting equation.

## Solution

Let $AB=50$ and $AC=41$ be the obliques, $AD$ be the perpendicular, $BD=10x, CD=3x$. Then $50^{2}-(10x)^{2}=AD^{2}=$ $41^{2}-9x^{2}$, or $91x^{2}=91 \cdot 9$, from which $x=3$, $BD=30, AD^{2}=50^{2}-30^{2}=40^{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_67482594d9a3488694ecg-25.jpg?height=321&width=500&top_left_y=2475&top_left_x=776)

Problem