Chapter 10.2, Problem 16PSC

To determine

To prove: that the diameter drawn through the intersection point bisects the angle formed by the chords, then the chords are congruent.

Answer to Problem 16PSC

  $AB\text{ is congruent }CD.$

Explanation of Solution

Given:

Concept used:

A perpendicular from the center to the chord bisects the chord.

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.

Calculation:

Consider the chords $AB\text{ and }CD$ interesting at $PEF$ is a diameter passing through $P$ , such that:

  $\begin{array}{l}\angle 1=\angle 2.\\ OM\perp ON.\end{array}$

  $OM$ and $ON$ are perpendiculars to the chord from center $O$ .

  $AB\cong CD.$

Consider right triangles $POM\text{ and }PON.$ :

  $\angle M=\angle N$ .

  $OM$ and $ON$ are perpendiculars to the chord from center $O$ .

  $\angle 1=\angle 2$ .

  $OP$ is a common side.

  $\Rightarrow \text{by }AAS$ congruence criterion $\Delta POM$ congruent to $\Delta PON$ .

congruent parts of congruent triangles are congruent.

  $\begin{array}{l}\Rightarrow OM=ON.\\ \Rightarrow AB=CD.\end{array}$

Hence, $AB\text{ is congruent }CD.$