[ The ratio in which the bisector divides the side. [The ratio of linear elements of similar triangles] A point on the hypotenuse, equidistant from both legs, divides the hypotenuse into segments of length 30 and 40. Find the legs of the triangle.

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The point equidistant from the sides of an angle lies on its bisector.

## Solution

Let $M$ be the specified point on the hypotenuse $AB$ of the right triangle $ABC$. Then $CM$ is the bisector of triangle $ABC$. By the property of the bisector of a triangle, $AC: BC = 4: 3$, which means this triangle is similar to a triangle with sides $3, 4, 5$. Since $70 = 14 \cdot 5$, then $BC = 14 \cdot 3 = 42$, $AC = 14 \cdot 4 = 56$.

![](https://cdn.mathpix.com/cropped/2024_05_06_9617f685748a8dfcd82ag-08.jpg?height=297&width=646&top_left_y=1741&top_left_x=706)

## Answer

42 and 56.