[ Properties of bisectors, concurrency ] [ The ratio in which the bisector divides the side $]$ [ Properties of bisectors, concurrency ]

In triangle $A B C$, points $M$ and $N$ are marked on sides $A B$ and $B C$ respectively, such that $B M=B N$. A line is drawn through point $M$ perpendicular to $B C$, and a line through point $N$ perpendicular to $A B$. These lines intersect at point $O$. The extension of segment $B O$ intersects side $A C$ at point $P$ and divides it into segments $A P=5$ and $P C=4$. Find the length of segment $B P$, given that $B C=6$.
The altitudes of triangle $B M N$, drawn from vertices $M$ and $N$, intersect at point $O$, which means that the altitude from vertex $B$ also passes through point $O$. Since triangle $B M N$ is isosceles, ray $B O$ is the angle bisector of $\angle M B N$, and $B P$ is the angle bisector of triangle $A B C$. By the angle bisector theorem, $\frac{B C}{A B} = \frac{C P}{A P}$, from which we find that

$$
A B = \frac{B C \cdot A P}{C P} = \frac{6 \cdot 5}{4} = \frac{15}{2}
$$

Using the formula for the angle bisector of a triangle,

$$
B P^2 = B C \cdot A B - C P \cdot A P = 6 \cdot \frac{15}{2} - 4 \cdot 5 = 45 - 20 = 25 \text{.}
$$

Therefore, $B P = 5$.

## Answer

5. 

| Problem 107711 Topics: | Quadrilateral Pyramid  Perpendicular and Oblique | Difficulty |
| :---: | :---: | :---: |
|  | [The larger angle lies opposite the larger side] | Grades: 10,11 |
|  | Inequalities with Angles |  |