Chapter 7.6, Problem 25WE

To determine

To find: The value of $MP$ .

Answer to Problem 25WE

  $MP=0.5$

Explanation of Solution

Given information:

The lengths of the sides of $\Delta ABC$ are $BC=12,CA=13,\text{ and }AB=14$ . M is the midpoint of $\overline{CA}$ , and P is the point where $\overline{CA}$ is cut by the bisector of $\angle B$ .

Calculation:

Since, M is the midpoint of $\overline{CA}$ , then, $AM=AC=\frac{13}{2}=6.5$

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

According to Triangle Angle-Bisector Theorem, if a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides.

Since, $BP$ bisects $\angle B$ ,

  $\begin{array}{l}\frac{AP}{PC}=\frac{AB}{BC}\\ \Rightarrow \frac{x}{13-x}=\frac{14}{12}\\ \Rightarrow x\times 12=14\times (13-x)[\text{cross multiplication]}\\ \Rightarrow 12x=14\times 13-14x\\ \Rightarrow 12x=182-14x\\ \Rightarrow 12x+14x=182\\ \Rightarrow 26x=182\\ \Rightarrow x=7\end{array}$

So, $AP=7$

Thus, $MP=AP-AM=7-6.5=0.5$