Chapter 9.5, Problem 15WE

To determine

To proveWhen the Centre O lies outside the angleATP

Explanation of Solution

Given Information:

The CentreO lies outside the angleATP as shown below:

Formulas Used:

• Theorem(1)

The measure of an inscribed angle is equal to half the measure of its intercepted arc

• Theorem(2)

The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc (The Theorem is proved when the Centre Olies on the chord)

• Arc addition postulate

The measure of the arc formed by two adjacent arcs is the sum of the measure of the two arcs

From the diagram given above,

Apply Theorem (2) to the chord ZT since the centreO lies on arc ZT

.......(1)

Apply Theorem (1) to the arc ZA

.......(2)

Apply angle addition postulate to the angle ZTP

......(3)

Apply arc addition postulate to the arc ZNT

.........(4)

Substitute (1) in Left Hand Side of (3) and (2) in Right Hand Side of (3)

∴ (3) $\Rightarrow$

  $\frac{1}{2}\stackrel{⌢}{ZNT}=\frac{1}{2}\stackrel{⌢}{ZA}+\angle ATP$…………(5)

Substitute (4) in Left Hand Side of (5)

  $\frac{1}{2}\left(\stackrel{⌢}{ZA}+\stackrel{⌢}{ANT}\right)=\frac{1}{2}\stackrel{⌢}{ZA}+\angle ATP$

  $\Rightarrow \frac{1}{2}\stackrel{⌢}{ZA}+\frac{1}{2}\stackrel{⌢}{ANT}=\frac{1}{2}\stackrel{⌢}{ZA}+\angle ATP$

  $\Rightarrow \frac{1}{2}\stackrel{⌢}{ANT}=\angle ATP$

Hence the proof.