Question

the circle below has center p.
the point $(x, y)$ is on the circle as shown.
(a) find the following.
center:
radius: units
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).
$\boldsymbol{\square}^2 + \boldsymbol{\square}^2 = \boldsymbol{\square}^2$

Explanation:

Step1: Identify center coordinates

From the graph, center \( P \) is at \((6, 8)\).

Step2: Identify radius length

The radius is labeled as 4 units.

Step3: Find \( a \) and \( b \) values

\( a \) is the horizontal distance from center to \((x,y)\): \( a = x - 6 \)
\( b \) is the vertical distance from center to \((x,y)\): \( b = y - 8 \)

Step4: Apply Pythagorean Theorem

For the right triangle, the legs are \( a \) and \( b \), hypotenuse is radius. Substitute \( a = x-6 \), \( b = y-8 \), radius = 4.

Answer:

Part (a)

Center: \((6, 8)\)
Radius: \(4\) units
Value of \(a\): \(x - 6\)
Value of \(b\): \(y - 8\)

Part (b)

\( (x-6)^2 + (y-8)^2 = 4^2 \)