Based on the diagram below, what is the measure of arc DC ? A 28 degrees B 68 degrees C 108 degrees D 112 degrees

Question

asked Nov 20, 2022 27.2k views

1 Answer

Xraynaud · Answer 1 · 2022-11-25T14:18:56+0000

Given:

A diagram of a circle O,
$arc(AB)=y^\circ, arc(CD)=(80+y)^\circ$ .

To find:

The measure of arc DC.

Solution:

According to the angle of intersecting chords theorem:

Angle at intersection of chords = Half of the sum of intercepted arcs.

Let
$\theta$ be the angle between the intersection of chords AC and BD.

$\theta +112^\circ=180^\circ$ (Linear pair)

$\theta =180^\circ-112^\circ$

$\theta =68^\circ$

Using the angle of intersecting chords theorem, we get

$\theta=(arc(AB)+arc(CD))/(2)$

$68^\circ =(y^\circ+(80+y)^\circ)/(2)$

$136^\circ =(80+2y)^\circ$

On simplification, we get

136=80+2y

136-80=2y

56=2y

Divide both sides by 2.

28=y

Now,

$arc(DC)=(80+y)^\circ$

$arc(DC)=(80+28)^\circ$

$arc(DC)=108^\circ$

Therefore, the correct option is C.