1. ## Triangles

Triangle ABC is isosceles with AC=BC=13 and AB=10. M and N are the feet of the perpendiculars from A and B to Bc and AC respectively. Find Bn.

2. Hello, Coach!

Triangle ABC is isosceles with $AC=BC=13,\;AB=10.$
$M \text{ and }N$ are the feet of the perpendiculars from $A \text{ and }B\text{ to }BC\text{ and }AC$, resp.
Find $BN.$
Code:
                C
*
/|\
/ | \
/  |  \
/   |   \
13 /    |    \ 13
/   12|     \
/      |      \
/       |       \
/        |        \
* - - - - + - - - - *
A    5    D    5    B
We see that $AD = 5,\;AC = 13.$
. . From Pythagorus: . $CD = 12$

Code:
                C
*
/|\
/ | \
/  |  \
/   |   \
N  /    |    \
*     |     \
/    * |      \
/       | *     \
/        |     *  \
* - - - - + - - - - *
A         D         B
We have similar right triangles: . $\Delta BNA \sim \Delta CDA$
. . (They share $\angle A$.)

Code:
                                            C
*
/|
/ |
/  |
/   |
N                        13  /    | 12
*                           /     |
/    *                      /      |
/         *                 /       |
/              *            /        |
* - - - - - - - - - *       * - - - - *
A        10         B       A    5    D
Hence, we have: . $\frac{BN}{10} \:=\:\frac{12}{13}$

Therefore: . $\boxed{BN \:=\:\frac{120}{13}}$