in the coordinate plane, the points (2,2) and (2,12) are the endpoints of a diameter of a circle. what is the equation of the circle in standard form?
the center is ... (2,7) ... and the radius is 5
(x - 2)^2 + (y - 7)^2 = 5^2
To find the equation of a circle given the coordinates of its endpoints, we can follow these steps:
Step 1: Find the center of the circle.
Since the diameter is a line segment that passes through the center of the circle, we can find the center by taking the average of the x-coordinates and the average of the y-coordinates of the endpoints.
x-coordinate of center = (x₁ + x₂) / 2
= (2 + 2) / 2
= 4 / 2
= 2
y-coordinate of center = (y₁ + y₂) / 2
= (2 + 12) / 2
= 14 / 2
= 7
Therefore, the center of the circle is (2, 7).
Step 2: Find the radius of the circle.
The radius of the circle is half the length of the diameter. We can find it by calculating the distance between the center and one of the endpoints of the diameter. Let's use (2, 2) as the endpoint.
radius = √((x₁ - x)² + (y₁ - y)²)
= √((2 - 2)² + (2 - 7)²)
= √(0² + (-5)²)
= √(0 + 25)
= √25
= 5
Therefore, the radius of the circle is 5.
Step 3: Write the equation in standard form.
The standard equation of a circle is (x - h)² + (y - k)² = r², where (h, k) represents the center coordinates and r represents the radius.
Plugging in the values, we have:
(x - 2)² + (y - 7)² = 5²
(x - 2)² + (y - 7)² = 25
Therefore, the equation of the circle in standard form is (x - 2)² + (y - 7)² = 25.