13. On a square $A B C D$ a line segment $B E$ is drawn such that the point $E$ lies on the side $C D$. The perimeter of triangle $B C E$ is three-quarters of the perimeter of the square $A B C D$. The ratio of lengths $C E: C D$ is $\lambda: 1$.
What is the value of $960 \times \lambda$ ?
SolUtion
720

Let the square's side have length 1 . The triangle's hypotenuse has length $\sqrt{1+\lambda^{2}}$, so the triangle's perimeter is $1+\lambda+\sqrt{1+\lambda^{2}}=3$. Therefore, $\sqrt{1+\lambda^{2}}=2-\lambda$ from which we obtain the equation $1+\lambda^{2}=4-4 \lambda+\lambda^{2}$.
Solving this equation we find $\lambda=\frac{3}{4}$, so $960 \lambda=720$.