Question

Based on the diagram below, what is the measure of arc DC ?

Answer 1

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

$\theta=\dfrac{arc(AB)+arc(CD)}{2}$

$68^\circ =\dfrac{y^\circ+(80+y)^\circ}{2}$

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

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.