[ Rectangles and Squares. Properties and Characteristics ]

In a plane, there is a square with vertices $A, B, C, D$ in sequence and a point $O$. It is known that $O B$ $=O D=13, O C=5 \sqrt{2}$, and that the area of the square is greater than 225. Find the length of the side of the square and determine whether point $O$ is inside or outside the square.
Point $O$ belongs to the line containing the diagonal $A C$ of the given square.

## Solution

Let $K$ be the center of the given square. Point $O$ is equidistant from vertices $B$ and $D$. Therefore, it lies on the perpendicular bisector of the diagonal $B D$, i.e., on the line containing the diagonal $A C$.

Let $a$ be the side of the given square. Since $S(A B C D)=a^{2}>225$, then $a>15$, and $B D=A C=a \sqrt{2}>15 \sqrt{2}$.

Therefore, $O C=5 \sqrt{2}+15 \sqrt{2} / 2=12.5 \sqrt{2}>13=O D$,

which is impossible, since $O K$ is the leg of the right triangle $O K D$ with hypotenuse $O D=13$. Therefore, point $O$ lies on the segment $C K$.

By the Pythagorean theorem from triangle $O K D$ we find:

$$
O D^{2}=O K^{2}+K D^{2}, \text{ or } 169=(a \sqrt{2} / 2-5 \sqrt{2})^{2}+(a \sqrt{2} / 2)^{2}.
$$

Solving this equation, we get that $a=17$.

![](https://cdn.mathpix.com/cropped/2024_05_06_76207e7feb35677daaaeg-50.jpg?height=375&width=440&top_left_y=2328&top_left_x=549)

Fig. 1.

![](https://cdn.mathpix.com/cropped/2024_05_06_76207e7feb35677daaaeg-50.jpg?height=297&width=346&top_left_y=2330&top_left_x=1165)

Fig. 2.

## Answer

17; inside.