Question

Type your answer and then click or tap Done.


The endpoints of a diameter of a circle are A(2, 1) and B(5, 5). Find the area of the circle in terms of π.

Answer 1

To find the area of the circle, we first need to determine its radius. Since we are given the endpoints of the diameter, we can use the distance formula to find the length of the diameter \(AB\).

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substitute the given points \(A(2, 1)\) and \(B(5, 5)\):

\[
AB = \sqrt{(5 - 2)^2 + (5 - 1)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

Therefore, the length of the diameter is 5. Since the radius \(r\) is half of the diameter, we have:

\[
r = \frac{5}{2} = 2.5
\]

The area \(A\) of a circle is given by the formula:

\[
A = \pi r^2
\]

Substitute the radius:

\[
A = \pi (2.5)^2 = \pi \left(\frac{5}{2}\right)^2 = \pi \left(\frac{25}{4}\right) = \frac{25\pi}{4}
\]

Therefore, the area of the circle is:

\[
\boxed{\frac{25\pi}{4}}
\]