A diameter of a circle has endpoints P(-7, 2) and Q(3, -8)
a. Find the center of the circle.
b. Find the radius. If your answer is not an integer, express it in radical form.
c. Write an equation for the circle.
a. To find the center of the circle, we need to find the midpoint of the line segment PQ, which is the center of the circle.
Midpoint formula:
(midpoint_x, midpoint_y) = ( (x1 + x2)/2 , (y1 + y2)/2 )
Using the coordinates given:
midpoint_x = (-7 + 3)/2 = -2
midpoint_y = (2 - 8)/2 = -3
Therefore, the center of the circle is (-2, -3).
b. The radius of the circle is half the length of the diameter. To find the length of the diameter we use the distance formula to find the distance between P and Q:
Distance formula:
distance = sqrt( (x2 - x1)^2 + (y2 - y1)^2 )
Using the coordinates given:
distance = sqrt( (3 - (-7))^2 + (-8 - 2)^2 ) = sqrt( 10^2 + (-10)^2 ) = sqrt(200)
Therefore, the radius is half the diameter which is:
radius = sqrt(200)/2 = 5sqrt(2)
c. The equation of a circle with center (h,k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Using the center and radius we found:
Equation of the circle:
(x - (-2))^2 + (y - (-3))^2 = (5sqrt(2))^2
Simplifying:
(x + 2)^2 + (y + 3)^2 = 50
For the radius, don't i need to solve 5√(2), or do i just leave it like that?
You can leave it as 5√(2) because that is the exact value of the radius. If you were asked to approximate the radius, then you would need to use a calculator to get a decimal approximation.
To find the center and radius of the circle, we need to use the midpoint formula and the distance formula.
a. Find the center of the circle:
The center of a circle is the midpoint of its diameter. We can find the midpoint using the coordinates of the endpoints of the diameter.
Midpoint formula:
The coordinates of the midpoint (x, y) of a line segment with endpoints (x1, y1) and (x2, y2) are given by:
x = (x1 + x2) / 2
y = (y1 + y2) / 2
Using the given coordinates:
x = (-7 + 3) / 2 = -4 / 2 = -2
y = (2 + -8) / 2 = -6 / 2 = -3
Therefore, the center of the circle is (-2, -3).
b. Find the radius:
The radius of a circle is equal to half the length of its diameter. We can find the length of the diameter using the distance formula.
Distance formula:
The distance between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the given coordinates:
d = sqrt((3 - -7)^2 + (-8 - 2)^2)
= sqrt((3 + 7)^2 + (-8 - 2)^2)
= sqrt(10^2 + -10^2)
= sqrt(100 + 100)
= sqrt(200)
Therefore, the radius of the circle is sqrt(200).
c. Write an equation for the circle:
The equation of a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Using the center (-2, -3) and radius sqrt(200):
(x - (-2))^2 + (y - (-3))^2 = (sqrt(200))^2
(x + 2)^2 + (y + 3)^2 = 200
Therefore, the equation of the circle is (x + 2)^2 + (y + 3)^2 = 200.