Question

N the circle below, if arc AC = 54 °, and arc BD = 158 °, find the measure of < BPD.

Answer 1

The measure of angle BPD in the circle is 106°, calculated by taking half the sum of the measures of intercepted arcs AC and BD.

Step-by-step explanation:

To find the measure of angle BPD in the circle, we need to understand that the angles formed by intersecting chords in a circle are equal to half the sum of the measures of the arcs they intercept. Here, angle BPD is an inscribed angle that intercepts arc BD and arc AC. Given that arc AC = 54° and arc BD = 158°, the measure of angle BPD will be equal to half the sum of these two arcs.

To calculate this, we use the formula:

angle BPD = ½(arc AC + arc BD)

Substitute the given values:

angle BPD = ½(54° + 158°) = ½(212°) = 106°

The measure of angle BPD is therefore 106°.

Answer 2

The relation between the arcs AC and BD with the angle BPD is given by:

`\begin{gathered} \angle BPD=\frac{arc\text{ AC}+arcBD}{2} \\ \angle BPD=(54\degree+158\degree)/(2) \\ \angle BPD=(212\degree)/(2) \\ \angle BPD=106\degree \end{gathered}`