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).
² + ² = ²

Explanation:

Step1: Determine Center of Circle

From the graph, the center \( P \) is at \( (7, 6) \) (by observing the coordinates of the center point on the grid).

Step2: Determine Radius

The radius is given as 4 units (from the line segment labeled with length 4 from center \( P \) to \( (x,y) \)).

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

• \( a \) is the horizontal distance from center \( P(7,6) \) to \( (x,y) \), so \( a = x - 7 \).
• \( b \) is the vertical distance from center \( P(7,6) \) to \( (x,y) \), so \( b = y - 6 \).

Step4: Apply Pythagorean Theorem

For the right triangle with legs \( a \) and \( b \), and hypotenuse (radius) 4, the Pythagorean theorem gives \( a^2 + b^2 = 4^2 \). Substituting \( a = x - 7 \) and \( b = y - 6 \), we get \( (x - 7)^2 + (y - 6)^2 = 16 \), or in terms of \( a \) and \( b \) directly, \( (x - 7)^2 + (y - 6)^2 = 4^2 \) which is \( (x - 7)^2 + (y - 6)^2 = 16 \). For part (a):

• Center: \( (7, 6) \)
• Radius: \( 4 \)
• \( a = x - 7 \), \( b = y - 6 \) (so value of \( a \) is \( x - 7 \), value of \( b \) is \( y - 6 \))

For part (b): Substituting into Pythagorean theorem, \( (x - 7)^2 + (y - 6)^2 = 4^2 \), so the equation is \( (x - 7)^2 + (y - 6)^2 = 16 \) (or in the blanks: \( (x - 7) \), \( (y - 6) \), \( 4 \)).

Answer:

(a)

• Center: \( (7, 6) \)
• Radius: \( 4 \)
• Value of \( a \): \( x - 7 \)
• Value of \( b \): \( y - 6 \)

(b) The equation is \( (x - 7)^2 + (y - 6)^2 = 4^2 \) (or filling the blanks: \( x - 7 \), \( y - 6 \), \( 4 \))