Chapter 9, Problem 22CR

a.

To determine

To find: The length of side BC.

Answer to Problem 22CR

The length of $\overline{BC}$ is 15.

Explanation of Solution

Given information:

  $\overline{BD}$ and $\overline{BE}$ trisect $\angle ABC$ .

Side $DE=4,$

Side $EC=5,$

Side $BD=12.$

Formula used:

The below theorem is used:

Angle bisector theorem states that “If a ray bisects an angle of a triangle, then it divides the opposite side of the triangle into segments that are proportional to the other two sides”.

  

  $\frac{a}{c}=\frac{b}{d}$

Calculation:

  

By Angle Bisector Theorem, we get

  $\begin{array}{l}\frac{BC}{EC}=\frac{BA}{AD}\\ \frac{BC}{5}=\frac{12}{4}\\ \therefore BC=15\end{array}$

b.

To determine

Tocalculate: The length of sideBE .

Answer to Problem 22CR

The length of $\overline{BE}$ is $4\sqrt{10}$ .

Explanation of Solution

Given information:

  $\overline{BD}$ and $\overline{BE}$ trisect $\angle ABC$ .

Side $DE=4,$

Side $EC=5,$

Side $BD=12.$

Formula used:

The below theorem is used:

Pythagoras theorem states that “In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

Calculation:

  

Note that $\Delta BDC$ is a triangle with sides $9,12$ and $15$ .

So it is right angle triangle with $\overline{BD}\perp \overline{AC}$ .

Side BE can be calculated by applying Pythagoras Theorem.

In right angle triangle BDE , we get

  $\begin{array}{l}{\left(BE\right)}^2={\left(BD\right)}^2+{\left(DE\right)}^2\\ {\left(BE\right)}^2={\left(12\right)}^2+{\left(4\right)}^2\\ {\left(BE\right)}^2=144+16\\ {\left(BE\right)}^2=160\\ \therefore BE=\sqrt{160}\\ \therefore BE=4\sqrt{10}\end{array}$

c.

To determine

To find: The length of sideAD .

Answer to Problem 22CR

The length of $\overline{AD}$ is 4.

Explanation of Solution

Given information:

  $\overline{BD}$ and $\overline{BE}$ trisect $\angle ABC$ .

Side $DE=4,$

Side $EC=5,$

Side $BD=12.$

Formula used:

The below property is used:

Two sides are congruent as there triangles are congruent by ASA congruence rule.

Calculation:

  

Two triangles are congruent by ASA congruence rule.

  $\Delta ABD\cong \Delta EBD$

Sides of congruent triangles are congruent.

  $\overline{AD}\cong \overline{DE}$

  $\overline{AD}=4$