Chapter 10.1, Problem 18PSB

To determine

To prove: that the line $\overrightarrow{AZ}$ passes through the point $P$ and $Z$ is the mid-point of the circle.

Answer to Problem 18PSB

  $AZ$ passes through the point $P$ .

Explanation of Solution

Given:

  

Concept used:

Condition of congruence triangle:

Side angle side $\left(SAS\right)$ : two side and one angle of one triangle is equivalent another triangle.

Side-side-side $\left(SSS\right)$ : all three side of the one triangle is equivalent to another triangle.

Angle side angle $\left(ASA\right)$ : two angle and side included between the angles of one triangle is equivalent to another two angle and side included between the angles of triangle.

Angle-angle side $\left(AAS\right)$:

two angle and non-included side between the angles of one triangle is equivalent to another two angle and non-included side between the angles of triangle.

Right angle hypotenuse side $\left(RHS\right)$: Hypotenuse and side of a right-angled triangle is equivalent to the hypotenuse and side of the second right angled triangle.

Perpendicular bisector of a chord passes through the center of the circle.

Calculation:

According to the given:

  $Z$ is the midpoint of $WX$ .

  $AZ$ is the median of the isosceles triangle it is known that for an isosceles triangle and the median is an Altitude.

Using $SSS\text{ or }SAS$ and then angles fin a straight line are supplementary.

  $AZ$ is the perpendicular to the bisector of $WX$ .

According to theorem the perpendicular bisector of a chord passes through the center of the circle.

Hence, $AZ$ passes through the point $P$ .