Chapter 8.5, Problem 21PSB

To determine

To find the length of the two sides.

Answer to Problem 21PSB

The length of AB is 10.5 and AC is 17.5

Explanation of Solution

Given information:

Perimeter of $\Delta ABC$ is 44.

Locate a point P on $\overline{BC}$ such that $\overrightarrow{AP}$ bisects $\angle BAC$

Formula used:

The below theorem is used:

Angle bisector theorem states that an angle bisector of a triangle will divide the opposite side into two segments that are proportional to the other two sides of the triangle.

Calculation:

  

  $P=44,BP=6,PC=10.$

Using Angle Bisector Theorem, we get

  $\begin{array}{l}\frac{BP}{BC}=\frac{AB}{AC}\\ \frac{6}{10}=\frac{AB}{AC}\\ 6AC=10AB\\ AB=44-BP-PC-AC\\ AB=44-10-6-17.5\\ AB=10.5\\ AB=28-AC\\ 6AC=10(28-AC)\\ 6AC=280-10AC\\ 16AC=280\\ \therefore AC=17.5\end{array}$