A diameter of a circle has endpoints P(-7, -4) and Q (3,2).
a. Find the center of the circle (hint: use midpoint formula).
b. Find the radius. If your answer is not and integer, express in radical form. (hint: use distance formula).
c. Write an equation for the circle.
Show work.
(This is 10th grade math, please don't make it advanced, shorten the steps).
AAAaannndd the bot gets it wrong yet again!
Use the distance to the center, not the other end of the diameter!
Distance = √((-2 - (-7))^2 + (-1 - (-4))^2) = √(5^2+3^2) = √34
so The equation for the circle is (x + 2)^2 + (y + 1)^2 = 34
Apologies for the error in the previous response, you are correct. The steps have been updated below:
a. Using midpoint formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint = ((-7 + 3)/2, (-4 + 2)/2)
Midpoint = (-2, -1)
The center of the circle is (-2, -1).
b. Using distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance = sqrt((-2 - (-7))^2 + (-1 - (-4))^2)
Distance = sqrt(5^2 + 3^2)
Distance = sqrt(34)
The radius of the circle is sqrt(34).
c. The general equation for a circle is (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center of the circle and r is the radius.
Plugging in the values from parts a and b:
(x - (-2))^2 + (y - (-1))^2 = (sqrt(34))^2
(x + 2)^2 + (y + 1)^2 = 34
The equation for the circle is (x + 2)^2 + (y + 1)^2 = 34.
a. To find the center of the circle, we can use the midpoint formula. The midpoint is the average of the x-coordinates and the average of the y-coordinates of the endpoints.
Midpoint formula: Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Here, the coordinates of P are (-7, -4) and the coordinates of Q are (3, 2).
x-coordinate of midpoint = (-7 + 3) / 2 = -4 / 2 = -2
y-coordinate of midpoint = (-4 + 2) / 2 = -2 / 2 = -1
Hence, the center of the circle is (-2, -1).
b. To find the radius of the circle, we can use the distance formula. The distance between the center of the circle and any of the endpoints will give us the radius.
Distance formula: Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Here, the coordinates of the center are (-2, -1) and the coordinates of Q are (3, 2).
Distance = √((3 - (-2))^2 + (2 - (-1))^2) = √((3 + 2)^2 + (2 + 1)^2) = √((5)^2 + (3)^2) = √(25 + 9) = √34
Hence, the radius of the circle is √34.
c. To write the equation of the circle, we can use the center-radius form of the equation.
Center-radius form: (x - h)^2 + (y - k)^2 = r^2
Here, the center of the circle is (-2, -1) and the radius is √34.
Equation of the circle: (x - (-2))^2 + (y - (-1))^2 = (√34)^2
(x + 2)^2 + (y + 1)^2 = 34
a. To find the center of the circle, we can use the midpoint formula. The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by the coordinates:
M = ((x1 + x2) / 2, (y1 + y2) / 2).
Here, the endpoints P(-7, -4) and Q(3, 2) give us the following coordinates:
x1 = -7, y1 = -4
x2 = 3, y2 = 2
Using the midpoint formula, we can calculate the x-coordinate and y-coordinate of the center as follows:
x-coordinate = (x1 + x2) / 2 = (-7 + 3) / 2 = -4 / 2 = -2.
y-coordinate = (y1 + y2) / 2 = (-4 + 2) / 2 = -2 / 2 = -1.
Therefore, the center of the circle is C(-2, -1).
b. To find the radius of the circle, we can use the distance formula. The distance between two points (x1, y1) and (x2, y2) is given by:
d = √[(x2 - x1)^2 + (y2 - y1)^2].
Here, we can calculate the distance between the endpoints P(-7, -4) and Q(3, 2) to determine the radius:
x1 = -7, y1 = -4
x2 = 3, y2 = 2
Using the distance formula, we can calculate the radius as follows:
radius = √[(3 - (-7))^2 + (2 - (-4))^2] = √[10^2 + 6^2] = √(100 + 36) = √136.
Therefore, the radius of the circle is √136.
c. To write the equation of the circle, we use the equation of a circle in standard form:
(x - h)^2 + (y - k)^2 = r^2,
where (h, k) represents the coordinates of the center, and r represents the radius.
From previous calculations, we know that the center is C(-2, -1), and the radius is √136. Plugging these values into the equation of a circle, we get:
(x - (-2))^2 + (y - (-1))^2 = (√136)^2.
(x + 2)^2 + (y + 1)^2 = 136.
Therefore, the equation of the circle is (x + 2)^2 + (y + 1)^2 = 136.
a. Using midpoint formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint = ((-7 + 3)/2, (-4 + 2)/2)
Midpoint = (-2, -1)
The center of the circle is (-2, -1).
b. Using distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance = sqrt((3 - (-7))^2 + (2 - (-4))^2)
Distance = sqrt(10^2 + 6^2)
Distance = sqrt(136)
The radius of the circle is sqrt(136).
c. The general equation for a circle is (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center of the circle and r is the radius.
Plugging in the values from parts a and b:
(x - (-2))^2 + (y - (-1))^2 = (sqrt(136))^2
(x + 2)^2 + (y + 1)^2 = 136
The equation for the circle is (x + 2)^2 + (y + 1)^2 = 136.