7.5. From a right triangle $A B C$ with legs $A C=3$ and $C B=7$, a square with vertex $C$ of the largest possible area needs to be cut out. What is the side length of such a square?
Answer: 2.1. Solution. Introduce a coordinate system: the origin - at vertex $C$, the $x$-axis - along $C A$, the $y$-axis - along $C B$. Then the hypotenuse lies on the line $y=7-\frac{7}{3} x$ (this follows from the meaning of the slope and the y-intercept for the graph of a line). The diagonal of the square can be written as the equation of the line $y=x$, and therefore the point of intersection of this line with the hypotenuse will be obtained by solving the given equations $x=7-\frac{7}{3} x \Leftrightarrow x=\frac{21}{10}$.

## 8th grade