162. Determine the segments into which the base $A C$ is divided by the perpendicular dropped from the opposite vertex $B$ of triangle $A B C$, if the sides of the triangle are given: $A B=15 ; B C=13$; $A C=14$.
162. Let $O C=x$ (Fig. 45), $O B^{2}=169-x^{2}$;

$$
A O^{2}=(14-x)^{2}=196-28 x+x^{2}
$$

$O B^{2}=225-A O^{2}=225-196+28 x-x^{2}=29+28 x-x^{2}$, therefore, $169-x^{2}=29+28 x-x^{2} ; \quad 140=28 x ; \quad x=5$.

$A O=9 ; \quad O B=\sqrt{169-25}=12$.

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-116.jpg?height=480&width=451&top_left_y=682&top_left_x=557)

Fig. 45.

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-116.jpg?height=674&width=229&top_left_y=491&top_left_x=1302)

Fig. 46.