Chapter 12.6, Problem 29E

(a)

To determine

To prove: That the $\Delta ABD$ is congruent to $\Delta CBD$ .

Answer to Problem 29E

It is proved that $\Delta ABD$ is congruent to $\Delta CBD$ .

Explanation of Solution

Given:

  $\begin{array}{c}\angle CDB\cong \angle ADB\\ \overline{DB}\perp \overline{AC}\end{array}$

Calculation:

The proof is as follows.

Statement	Reasons
$\begin{array}{c}\angle CDB\cong \angle ADB\\ \overline{DB}\perp \overline{AC}\end{array}$	Given
$\angle ABD$ and $\angle CBD$ are right angles	Definition of perpendicular lines
$\angle ABD\cong \angle CBD$	Right Angles Congruence Theorem
$\overline{BD}\cong \overline{BD}$	Reflexive Property of Congruence
$\Delta ABD\cong \Delta CBD$	ASA Congruence Theorem

Therefore, it is proved that $\Delta ABD$ is congruent to $\Delta CBD$ .

(b)

To determine

To verify: That the $\Delta ACD$ is isosceles.

Answer to Problem 29E

It is verified that $\Delta ACD$ is isosceles.

Explanation of Solution

Given:

  $\begin{array}{c}\angle CDB\cong \angle ADB\\ \overline{DB}\perp \overline{AC}\end{array}$

Calculation:

Since $\Delta ABD\cong \Delta CBD$ and corresponding parts of congruent triangles are congruent.

Therefore, $\overline{AD}\cong \overline{CD}$ , that is $\Delta ACD$ is isosceles.

(c)

To determine

To find: Whether moving away from the mirror have any effect on the amount of his or her reflection person sees.

Answer to Problem 29E

The answer is no.

Explanation of Solution

Given:

The $\Delta ACD$ is

Calculation:

The answer is no, because $\Delta ACD$ isosceles. The girl sees her toes in the bottom of the mirror. This remains true as she moves backwards because $\Delta ACD$ remains isosceles.