Question

the circle below has center p. the point (x, y) is on the circle as shown. (a) find the following. radius: units center: ( , ) value of a: select value of b: select (b) use the pythagorean theorem to write an equation relating the side lengths of the right triangle. write your answer in terms of x and y (with no other letters). ()² + ()² = ()²

Explanation:

Step1: Determine radius

By observing the graph, the length of the line - segment from the center \(P\) to the point \((x,y)\) is given as \(2\) units. So the radius \(r = 2\) units.

Step2: Find center coordinates

From the graph, the center \(P\) is at the point \((3,7)\).

Step3: Calculate \(a\) and \(b\)

The \(x\) - coordinate of the center is \(3\) and the \(x\) - coordinate of the point on the circle is \(x\). The horizontal distance \(a=x - 3\). The \(y\) - coordinate of the center is \(7\) and the \(y\) - coordinate of the point on the circle is \(y\). The vertical distance \(b=y - 7\).

Step4: Apply Pythagorean theorem

In the right - triangle formed, the two legs have lengths \(a=x - 3\) and \(b=y - 7\), and the hypotenuse is the radius \(r = 2\). According to the Pythagorean theorem \(a^{2}+b^{2}=r^{2}\), so \((x - 3)^{2}+(y - 7)^{2}=2^{2}\).

Answer:

(a)
Radius: \(2\) units
Center: \((3,7)\)
Value of \(a\): \(x - 3\)
Value of \(b\): \(y - 7\)
(b) \((x - 3)^{2}+(y - 7)^{2}=2^{2}\)