Question

What are the coordinates of the centre of the circle ?

Answer 1

We are given the coordinates of the two endpoints of a circle's diameter. We need to find the coordinates of the center of the circle.

The center of a circle is the midpoint of its diameter. The midpoint formula for two points (x₁, y₁) and (x₂, y₂) is:
Midpoint = ( (x₁ + x₂)/(2), (y₁ + y₂)/(2) ) The endpoints of the diameter are (x₁, y₁) = (4, 3) and (x₂, y₂) = (8, 5).
Let the coordinates of the center be (x, y). Using the midpoint formula:
x = (x₁ + x₂)/(2) = (4 + 8)/(2) = (12)/(2) = 6 y = (y₁ + y₂)/(2) = (3 + 5)/(2) = (8)/(2) = 4 So, the coordinates of the center of the circle are (6, 4).

The coordinates of the centre of the circle are (6, 4).

Answer 2

We are considering the same circle from the previous question, so we know its center. We are given one endpoint of a new diameter and we need to find the other endpoint.

The center of the circle is the midpoint of the diameter. Let the known endpoint be (x₁, y₁) and the unknown endpoint be (x₂, y₂). Let the center be (xc, yc). The midpoint formula is:
xc = (x₁ + x₂)/(2) and yc = (y₁ + y₂)/(2) We can rearrange this to solve for the unknown endpoint coordinates:
x₂ = 2xc - x₁ and y₂ = 2yc - y₁ From part (i), the center of the circle is (xc, yc) = (6, 4).
The given endpoint of the new diameter is (x₁, y₁) = (5, 2).
Let the other endpoint be (x₂, y₂).
Using the rearranged midpoint formula:
x₂ = 2(6) - 5 = 12 - 5 = 7 y₂ = 2(4) - 2 = 8 - 2 = 6 So, the coordinates of the other end of the diameter are (7, 6).

The coordinates of the other end of that diameter are (7, 6).