Question

the circle below has center p. the point (x, y) is on the circle as shown. (a) find the following. radius: 2 units center: (3, 7) value of a: select ^ value of b: x - 4 4 - x (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). (x - 3)^2 + y - 4 = (2)^2 4 - y y - 7 7 - x x - 7 7 - y

Explanation:

Step1: Find the value of \(a\)

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

Step2: Find the value of \(b\)

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

Step3: Write the Pythagorean - equation

For a right - triangle formed with the radius as the hypotenuse, by the Pythagorean theorem \(a^{2}+b^{2}=r^{2}\). Substituting \(a=x - 3\), \(b=y - 7\), and \(r = 2\), we get \((x - 3)^{2}+(y - 7)^{2}=4\).

Answer:

(a) Value of \(a\): \(x - 3\)
Value of \(b\): \(y - 7\)
(b) \((x - 3)^{2}+(y - 7)^{2}=4\)