Chapter 10.6, Problem 16PSB

To determine

Find the value of the given arc using given data.

Answer to Problem 16PSB

The value of $arcAE={86}^{°}$ .

Explanation of Solution

Given:

The given diagram is

  

$O$ and $P$ are the center of the big circle and small circle respectively. $\overline{AC}$ and $\overline{CE}$ are the tangents of small circle at point $B$ and $D$ . The given arc angle $DFB={223}^{°}$ . The given arc is $AE$ .

Calculation:

The one rotation circle angle $={360}^{°}$

In diagram, the give $marcDFB={223}^{°}$

  $marcDB={360}^{°}-{223}^{°}={137}^{°}$

Use tangent-tangent angle theorem.

  $m\angle C={180}^{°}-{137}^{°}={43}^{°}$

If an inscribed angle composed by two chords which vertex is on the circle, then the $marc=2\times \text{vertex}\text{angle}$ .

  $marcAE=2m\angle C=2\cdot {43}^{°}={86}^{°}$

Hence the value of $arcAE={86}^{°}$ .