Question

A circle has a center at  (−4,−1)  and a point at  (2,−1.5) . What is the length of the diameter of the circle? Round your answer to the nearest tenth.

Answer 1

To find the length of the diameter of the circle, we first need to determine its radius. The radius is the distance from the center of the circle to any point on the circle.

The center of the circle is at \((-4, -1)\) and a point on the circle is at \((2, -1.5)\). We can use the distance formula to calculate the distance \(d\) between these two points:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Assigning our points:
- \((x_1, y_1) = (-4, -1)\)
- \((x_2, y_2) = (2, -1.5)\)

Now we can plug in these values into the distance formula:

\[
d = \sqrt{(2 - (-4))^2 + (-1.5 - (-1))^2}
\]
\[
= \sqrt{(2 + 4)^2 + (-1.5 + 1)^2}
\]
\[
= \sqrt{(6)^2 + (-0.5)^2}
\]
\[
= \sqrt{36 + 0.25}
\]
\[
= \sqrt{36.25}
\]
\[
= 6.020408163 (approximately)
\]

This distance \(d\) is the radius of the circle. To find the diameter \(D\), we use the formula:

\[
D = 2 \times \text{radius}
\]
\[
D = 2 \times 6.020408163 \approx 12.04081633
\]

Rounding this to the nearest tenth gives us:

\[
D \approx 12.0
\]

Thus, the length of the diameter of the circle is approximately \( \boxed{12.0} \).