[ Pythagorean Theorem (direct and inverse) ]  [ Isosceles, inscribed, and circumscribed trapezoids ]

In an isosceles trapezoid \(ABCD\), the base \(AD\) is 4, and the base \(BC\) is 3. Points \(M\) and \(N\) lie on the diagonal \(BD\), with point \(M\) located between points \(B\) and \(N\), and segments \(AM\) and \(CN\) are perpendicular to the diagonal \(BD\). Find \(CN\) if \(BM: DN = 2: 3\).
Using the similarity of triangles and the Pythagorean theorem, find the ratio $B M: M O: O N: N D$.

## Solution

Let $O$ be the point of intersection of the diagonals of trapezoid $A B C D$. From the similarity of triangles $B O C$ and $D O A$, it follows that $B O: O D=B C: A D=3: 4$, and from the similarity of triangles $C O N$ and $A O M-O N: O M=C O: O A=3: 4$.

Let $B M=2 x, D N=3 x, O N=3 y, O M=4 y$. Since $B O: O D=3: 4$, then $4(2 x+4 y)=3(3 x+3 y)$, from which $x=7 y$. Therefore, $B O=O C=2 x+4 y=18 y$,

$B N=21 y$.

We have $9=B C^{2}=B N^{2}+C N^{2}=B N^{2}+O C^{2}-O N^{2}=y^{2}\left(21^{2}+18^{2}-3^{2}\right)=9 y^{2}\left(7^{2}+6^{2}-1\right)=9 \cdot 84 y^{2}$.

From this, $y^{2}=1 / 84$. Therefore, $C N^{2}=B C^{2}-21^{2} y^{2}=9-21 / 4=15 / 4$.

![](https://cdn.mathpix.com/cropped/2024_05_06_6ed8f1205731b80a0039g-39.jpg?height=291&width=595&top_left_y=2633&top_left_x=726)