Question

the figure shows a circle of radius 2 along with four labeled points in the xy - plane. the measure of angle cob is equal to the measure of angle aob. what are the coordinates of point b?

Explanation:

Step1: Determine the angle for point A

Point A is at \((0, -2)\), so the angle for OA is \(270^\circ\) or \(\frac{3\pi}{2}\) radians.

Step2: Analyze the angle relationship

Given \(\angle COB=\angle AOB\). Point C is at \((2, 0)\) (angle \(0^\circ\)). Let the angle of OB with the positive x - axis be \(\theta\). The angle of OA with the positive x - axis is \(270^\circ\), so the angle between OA and the negative y - axis is \(0^\circ\), and the angle between OC (x - axis) and OA is \(90^\circ\) (since OA is along the negative y - axis). Let \(\angle AOB = \angle COB=\alpha\). The total angle from OC to OA is \(90^\circ\), so \(2\alpha=90^\circ\), then \(\alpha = 45^\circ\). So the angle of OB with the positive x - axis is \(- 45^\circ\) (or \(315^\circ\)) because it is below the x - axis.

Step3: Use the parametric equations of a circle

The parametric equations of a circle with radius \(r\) centered at the origin are \(x = r\cos\theta\) and \(y=r\sin\theta\), where \(r = 2\) and \(\theta=-45^\circ=\frac{- \pi}{4}\) radians.

We know that \(\cos(-45^\circ)=\cos(45^\circ)=\frac{\sqrt{2}}{2}\) and \(\sin(-45^\circ)=-\sin(45^\circ)=-\frac{\sqrt{2}}{2}\)

Step4: Calculate the coordinates of B

For \(x\) - coordinate: \(x = 2\times\cos(-45^\circ)=2\times\frac{\sqrt{2}}{2}=\sqrt{2}\)

For \(y\) - coordinate: \(y = 2\times\sin(-45^\circ)=2\times(-\frac{\sqrt{2}}{2})=-\sqrt{2}\)

Answer:

The coordinates of point B are \((\sqrt{2},-\sqrt{2})\)