4. Let $A B C$ be a right triangle. Point $D$ is the midpoint of the hypotenuse $\overline{A B}$, $F$ is the midpoint of side $\overline{A C}$, $E$ is the midpoint of $\overline{C F}$, and $G$ is the midpoint of $\overline{F A}$. Segment $\overline{C D}$ intersects $\overline{B E}$, $\overline{B F}$, and $\overline{B G}$ at points $P$, $Q$, and $R$ respectively. What is the ratio $\frac{|P Q|}{|Q R|}$?
4. Set up a coordinate system so that its origin is at point $C$, and sides $\overline{C A}$ and $\overline{C B}$ lie on the $x$ and $y$ coordinate axes. Now, $C(0,0)$, $A(b, 0)$, $B(0, a)$, $D\left(\frac{b}{2}, \frac{a}{2}\right)$, $E\left(\frac{b}{4}, 0\right)$, $F\left(\frac{b}{2}, 0\right)$, $G\left(\frac{3 b}{4}, 0\right)$.

![](https://cdn.mathpix.com/cropped/2024_06_03_843fcc95a1719fe2319dg-10.jpg?height=282&width=391&top_left_y=1518&top_left_x=587)

.

5 points

The equations of the lines $C D, B E, B F$ and $B G$ are

$\begin{array}{lllllll}C D & \ldots & y=\frac{a}{b} x, & B E & \ldots & y=-\frac{4 a}{b} x+a, \\ B F & \ldots & y=-\frac{2 a}{b} x+a, & B G & \ldots & y=-\frac{4 a}{3 b} x+a . & 10 \text { points }\end{array}$

Next, determine the points $P, Q$ and $R$.

$P=C D \cap B E \quad \ldots \quad P\left(\frac{b}{5}, \frac{a}{5}\right)$,

$Q=C D \cap B F \quad \ldots \quad Q\left(\frac{b}{3}, \frac{a}{3}\right)$,

$R=C D \cap B G \quad \ldots \quad R\left(\frac{3 b}{7}, \frac{3 a}{7}\right)$.

5 points

Now, $|P Q|=\frac{2}{15} \sqrt{a^{2}+b^{2}}, \quad|Q R|=\frac{2}{21} \sqrt{a^{2}+b^{2}}$.

The desired ratio is $\quad \frac{|P Q|}{|Q R|}=\frac{7}{5}$.

5 points

## MINISTRY OF EDUCATION AND SPORT OF THE REPUBLIC OF CROATIA

## CROATIAN MATHEMATICAL SOCIETY

## MATHEMATICS