Question

Based on the diagram Below,what is the measure of arc DC? Given 80+y ,and 112.

Answer 1

In order to determine the value of arc DC we need first to determine the angle between the two chords. To do that we will use the fact that two times this angle plus two times the angle of 112 must be equal to 360, that is:

`2x+2(112)=360`

Simplifying:

`\begin{gathered} x+112=(360)/(2) \\ x+112=180 \end{gathered}`

Subtracting 112 to both sides:

`\begin{gathered} x=180-112 \\ x=68 \end{gathered}`

Now we will use the following relationship between arcs and angles:

`2x=\text{arcDC}+\text{arcAB}`

Replacing the values:

`2(68)=(80+y)+y`

Simplifying:

`136=80+2y`

Now we solve for "y" first by subtracting 80 to both sides:

`\begin{gathered} 136-80=2y \\ 56=2y \end{gathered}`

Dividing both sides by 2:

`\begin{gathered} (56)/(2)=y \\ 28=y \end{gathered}`

Now we replace the value of "y" in the expression for arc DC:

`\begin{gathered} \text{arcDC}=80+y \\ \text{arcDC}=80+28 \\ \text{ArcDC}=108 \end{gathered}`

Therefore, ArcDC measure 108.