Question

the circle below has center p.
the point (x, y) is on the circle as shown.
(a) find the following.
radius: \boxed{} units
center: (\boxed{}, \boxed{})
value of a: select
value of b: y - 5
4 - y
(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).
(\boxed{})^2 + (\boxed{})^2
x - 4
4 - x
y - 4
5 - y

Explanation:

Step1: Determine the radius

From the diagram, the length of the radius (the hypotenuse of the right triangle) is given as 2 units. So the radius is 2.

Step2: Find the center coordinates

Looking at the graph, the center \( P \) has coordinates. From the horizontal and vertical positions, the x - coordinate of \( P \) is 4 and the y - coordinate is 5. So the center is \( (4,5) \).

Step3: Find the value of \( a \)

The horizontal leg of the right triangle (length \( a \)) is the difference between the x - coordinate of the point \( (x,y) \) and the x - coordinate of the center. So \( a=x - 4 \) (since \( x>4 \), the length is \( x - 4 \)).

Step4: Find the value of \( b \)

The vertical leg of the right triangle (length \( b \)) is the difference between the y - coordinate of the point \( (x,y) \) and the y - coordinate of the center. Since the center's y - coordinate is 5, \( b = y - 5 \).

Step5: Use the Pythagorean theorem

The Pythagorean theorem states that for a right triangle with legs \( a \) and \( b \) and hypotenuse \( c \), \( a^{2}+b^{2}=c^{2} \). Here, \( a=x - 4 \), \( b = y - 5 \), and \( c = 2 \) (the radius). So the equation is \( (x - 4)^{2}+(y - 5)^{2}=2^{2}=4 \).

Part (a)

• Radius: 2 units (from the diagram, the length of the radius is given as 2)
• Center: \( (4,5) \) (by looking at the position of center \( P \) on the coordinate plane)
• Value of \( a \): \( x - 4 \) (horizontal distance between \( (x,y) \) and center \( (4,5) \))
• Value of \( b \): \( y - 5 \) (vertical distance between \( (x,y) \) and center \( (4,5) \))

Part (b)

Using Pythagorean theorem \( a^{2}+b^{2}=c^{2} \), substituting \( a=x - 4 \), \( b = y - 5 \), and \( c = 2 \), we get \( (x - 4)^{2}+(y - 5)^{2}=4 \)

Answer:

Part (a)

• Radius: \(\boldsymbol{2}\) units
• Center: \(\boldsymbol{(4, 5)}\)
• Value of \(a\): \(\boldsymbol{x - 4}\)
• Value of \(b\): \(\boldsymbol{y - 5}\)

Part (b)

The equation is \(\boldsymbol{(x - 4)^{2}+(y - 5)^{2}=4}\)