Question

1. on circle c shown below, the points g, u, i, t, a, and r are spaced evenly. answer the following questions.

(a) what is the measure of ∠gcu? show how you arrived at this number.
(b) what angle of rotation about point c would be needed to map point g to point a if the rotation is counterclockwise?
(c) if point u was rotated clockwise about point c by an angle of 240°, at what point would it land?

Explanation:

(a)

Step1: Calculate the measure of one central angle

Since there are 6 evenly - spaced points on the circle, the measure of one central angle is \(\frac{360^{\circ}}{6}\).
\(\frac{360^{\circ}}{6}=60^{\circ}\)

Step2: Determine the measure of \(\angle GCU\)

\(\angle GCU\) is one of the central angles formed by two adjacent points.

Step1: Count the number of intervals from \(G\) to \(A\) counter - clockwise

From \(G\) to \(A\) counter - clockwise, there are 4 intervals.

Step2: Calculate the rotation angle

Since each interval is \(60^{\circ}\), the rotation angle is \(4\times60^{\circ}\).
\(4\times60^{\circ} = 240^{\circ}\)

Step1: Analyze the clockwise rotation of \(240^{\circ}\)

A full - circle rotation is \(360^{\circ}\). A \(240^{\circ}\) clockwise rotation is equivalent to a \(360^{\circ}-240^{\circ}=120^{\circ}\) counter - clockwise rotation.
Since each interval is \(60^{\circ}\), \(120^{\circ}\div60^{\circ} = 2\) intervals counter - clockwise from \(U\).

Answer:

\(60^{\circ}\)

(b)