B2. Let there be a quadrilateral $A B C D$ with right angles at vertices $A$ and $C$. The orthogonal projections of vertices $D$ and $B$ onto segment $A C$ are denoted by $E$ and $F$ respectively. What is the length of segment $B F$, if $|A E|=3$, $|D E|=5$, and $|C E|=7$?
B2.

![](https://cdn.mathpix.com/cropped/2024_06_07_95af3fbbcea2fb458cf3g-10.jpg?height=551&width=589&top_left_y=1575&top_left_x=745)

Since the angles at vertices $A$ and $C$ of quadrilateral $ABCD$ are right angles, points $A$ and $C$ lie on a circle with diameter $BD$ according to Thales' theorem. Therefore, quadrilateral $ABCD$ is cyclic. By the Pythagorean theorem, $|AD| = \sqrt{|AE|^2 + |DE|^2} = \sqrt{9 + 25} = \sqrt{34}$ and $|CD| = \sqrt{|CE|^2 + |DE|^2} = \sqrt{49 + 25} = \sqrt{74}$. Since quadrilateral $ABCD$ is cyclic, $\angle CBD = \angle CAD$. Therefore, triangles $BCD$ and $AED$ are similar, as they have two pairs of congruent angles. From this, it follows that $\frac{|BD|}{|CD|} = \frac{|AD|}{|ED|}$, and thus $|BD| = \frac{|CD| \cdot |AD|}{|ED|} = \frac{\sqrt{74} \cdot \sqrt{34}}{5}$. By the Pythagorean theorem,

$$
|AB| = \sqrt{|BD|^2 - |AD|^2} = \sqrt{\frac{74 \cdot 34}{25} - 34} = \sqrt{\frac{74 \cdot 34 - 25 \cdot 34}{25}} = \sqrt{\frac{49 \cdot 34}{25}} = \frac{7}{5} \sqrt{34}
$$

and

$$
|BC| = \sqrt{|BD|^2 - |CD|^2} = \sqrt{\frac{74 \cdot 34}{25} - 74} = \sqrt{\frac{74 \cdot 34 - 25 \cdot 74}{25}} = \sqrt{\frac{9 \cdot 74}{25}} = \frac{3}{5} \sqrt{74}
$$

Triangles $ABF$ and $DBC$ are also similar, as due to the cyclic nature of quadrilateral $ABCD$, $\angle BAC = \angle BDC$. Therefore, $\frac{|BF|}{|AB|} = \frac{|BC|}{|DB|}$, from which it follows that

$$
|BF| = \frac{|AB| \cdot |BC|}{|DB|} = \frac{\frac{7}{5} \sqrt{34} \cdot \frac{3}{5} \sqrt{74}}{\frac{\sqrt{74} \cdot \sqrt{34}}{5}} = \frac{21}{5}
$$

Discovery that quadrilateral $ABCD$ is cyclic .1 point

Application of the Pythagorean theorem to calculate $|AD|$ or $|CD|$ 1 point

Discovery that triangles $BCD$ and $AED$ are similar 1 point Calculation of $|BD|$ .1 point

Calculation of $|AB|$ or $|BC|$ .1 point

Discovery that triangles $ABF$ and $DBC$ are similar .1 point

Correctly calculated length of $|BF|$ 1 point