Triangles

Hello, Coach!

Quote:

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