Problem 8.7. Given a right isosceles triangle $ABC$ with a right angle at $A$. A square $KLMN$ is positioned as shown in the figure: points $K, L, N$ lie on sides $AB, BC, AC$ respectively, and point $M$ is located inside triangle $ABC$.

Find the length of segment $AC$, if it is known that $AK=7, AN=3$.

![](https://cdn.mathpix.com/cropped/2024_05_06_1281ceb10571c66f6e2bg-08.jpg?height=496&width=495&top_left_y=535&top_left_x=479)
Answer: 17.

![](https://cdn.mathpix.com/cropped/2024_05_06_1281ceb10571c66f6e2bg-08.jpg?height=492&width=495&top_left_y=1157&top_left_x=479)

Fig. 2: to the solution of problem 8.7

Solution. Mark a point $H$ on the segment $B K$ such that $L H \perp B K$ (Fig. 2). Triangle $B H L$ is a right isosceles triangle, $H B=H L$. Note that $\angle L K H=90^{\circ}-$
$\angle A K N=\angle A N K$, and also $K L=K N$. Right triangles $L K H$ and $K N A$ are equal by an acute angle and the hypotenuse, so $K H=A N=3$ and $L H=A K=7$. Therefore,

$$
A C=A B=A K+K H+H B=7+3+7=17
$$