Chapter 4.7, Problem 7CE

(a)

To determine

Draw $\overline{XY}$ .Label its midpoint Q.Select a point P equidistant from X and Y. Draw $\overline{PX}$ , $\overline{PY}$ , and $\overline{PQ}$ .

Explanation of Solution

Calculation:

The diagram is :

Draw $\overline{XY}$ .Label its midpoint Q.

  

A point P equidistant from X and Y . Draw $\overline{PX}$ , $\overline{PY}$ , and $\overline{PQ}$ .

:

  

(b)

To determine

Draw $\overline{XY}$ .Label its midpoint Q.Find the postulate that justifies the statement $\Delta PQX\cong \Delta PQY$ .

Answer to Problem 7CE

SSS Postulate.

Explanation of Solution

Given:

From part (a) , we have:

Q is the midpoint of $\overline{XY}$ and P is equidistant from X and Y.

Calculation:

  

Consider $\Delta PQX\text{ and }\Delta \text{PQY}$ :

  $\overline{XQ}\cong \overline{QY}$

  $\overline{PX}\cong \overline{PY}$

  $\overline{PQ}\cong \overline{PQ}$

So, by SSS Postulate , $\Delta PQX\cong \Delta PQY$

(c)

To determine

Find the justification of $\angle PQX\cong \angle PQY$ .

Answer to Problem 7CE

Because $\Delta PQX\cong \Delta PQY$

Explanation of Solution

Given:

From part(b) , $\Delta PQX\cong \Delta PQY$

Calculation:

Since , $\Delta PQX\cong \Delta PQY$

So, we have $\angle PQX\cong \angle PQY$

(d)

To determine

Find the justification of $\overleftrightarrow{PQ}\perp \overline{XY}$ .

Answer to Problem 7CE

In an isosceles triangle , the median , altitude and the angle bisector of the vertex , all are congruent.

Explanation of Solution

Given:

  

From part (a) , we have:

Q is the midpoint of $\overline{XY}$ and P is equidistant from X and Y.

Calculation:

  

  $\overline{PX}\cong \overline{PY}$ , so , the $\Delta PXY$ is isosceles and we know that in an isosceles triangle , the median , altitude and the angle bisector of the vertex , all are congruent.

Since $\overline{PQ}$ is the median, so it is also an altitude.

Hence, $\overleftrightarrow{PQ}\perp \overline{XY}$

(e)

To determine

Find the name of $\overleftrightarrow{PQ}$ that best describes the relationship between $\overleftrightarrow{PQ}$ and $\overline{XY}$ .

Answer to Problem 7CE

  $\overleftrightarrow{PQ}$ is the perpendicular bisector of $\overline{XY}$ .

Explanation of Solution

Calculation:

From part (c) and (a) , we have:

  $\overleftrightarrow{PQ}$ is the median and $\overleftrightarrow{PQ}\perp \overline{XY}$

So, $\overleftrightarrow{PQ}$ is the perpendicular bisector of $\overline{XY}$ .