[ Signs of Similarity

Find the length of the chord if the radius $r$ of the circle and the distance $a$ from one end of the chord to the tangent drawn through the other end are given.

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Drop a perpendicular from the center of the circle to the given chord and consider the similar triangles formed.

## Solution

Let $O$ be the center of the given circle, $M$ be the foot of the perpendicular dropped from the end $B$ of the chord $A B$ to the tangent to the circle at point $A$, and $K$ be the midpoint of $A B$.

Since triangles $A K O$ and $B M A$ are similar, we have $A K: A O = M B: A B$, or $A B: 2 r = a: A B$. Therefore, $A B = \sqrt{2 a r}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_36c4956010bd91c4b4eeg-21.jpg?height=349&width=483&top_left_y=437&top_left_x=793)

## Answer

$\sqrt{2 a r}$.