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Here's #3 . . .

3) Lengths of the sides of triangle $\displaystyle ABC$ are: $\displaystyle BC=12,\:CA= 13,\:AB = 14$

If $\displaystyle M$ is the midppoint of $\displaystyle CA$ and $\displaystyle P$ is point where $\displaystyle CA$ is cut by the bisector of $\displaystyle \angle B$,

find the length of MP. Code:

A
o
* *
* * 13-x
* *
* *
14 * * P
* o
* * * x
* * *
* * *
B o * * * * * * * * * o C
12

$\displaystyle BP$ is the bisector of $\displaystyle \angle B:\;\angle ABP = \angle PBC$

Let $\displaystyle x = PC$, then $\displaystyle AP = 13-x$

Theorem: An angle bisector divides the opposite side into two segments

. . . . . . . proportional to the two adjacent sides.

So we have: .$\displaystyle \frac{x}{13-x} = \frac{12}{14}\quad\Rightarrow\quad x \,=\,PC \,= \,6$

Since $\displaystyle M$ is the midpoint of $\displaystyle CA,\;MC = 6.5$

Therefore: .$\displaystyle MP \:=\:MC - PC \:=\:6.5 - 6 \:=\:\boxed{0.5}$