Question

point f lies on a unit circle in the xy - plane and has coordinates (0, 0). point h also lies on the circle and has coordinates (1, 0). point g is the center of the circle and has coordinates (-1, y), where y is a constant. which of the following could be the positive measure of angle fgh, in radians? a $\frac{37pi}{2}$ b $\frac{39pi}{2}$ c 34$pi$ d 35$pi$

Explanation:

Step1: Recall properties of unit - circle

The center of the unit - circle is at the origin \((0,0)\). Given \(G\) is the center of the circle with coordinates \((0,0)\), \(H=(1,0)\) and \(F=( - 1,y)\). The angle between two points on a unit - circle can be found using the arc - length formula \(s = r\theta\) (since \(r = 1\) for a unit - circle, \(s=\theta\)). The angle between two points \((x_1,y_1)\) and \((x_2,y_2)\) on a unit - circle centered at the origin can also be found by considering the rotation from one point to the other. The point \(H=(1,0)\) corresponds to an angle of \(0\) radians and the point \(F=( - 1,y)\) corresponds to an angle of \(\pi\) radians plus some multiple of \(2\pi\) radians.

Step2: Calculate the angle

The angle \(\theta\) from \(H=(1,0)\) to \(F=( - 1,y)\) is \(\pi\) radians for the shortest path. But we are looking for positive measures. We know that the general form of an angle in standard position is \(\theta+2k\pi\), \(k\in\mathbb{Z}\).
If we start from the positive \(x\) - axis (\(H=(1,0)\)) and move to the point \(F=( - 1,y)\), the positive measure of the angle is \(\pi+2k\pi\). When \(k = 17\), \(\theta=\pi + 34\pi=35\pi\).

Answer:

D. \(35\pi\)