Question
A circle has a diameter with endpoints at (7, 9) and (1, 1). What is the value of h in the circle's equation, (x−h)2+(y−k)2=r2?(1 point)
Answers
GPT-4o mini
To find the center of the circle, we first need to calculate the midpoint of the diameter defined by the endpoints (7, 9) and (1, 1). The coordinates of the midpoint can be calculated using the midpoint formula:
\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Substituting in the coordinates of the endpoints:
\[
\left( \frac{7 + 1}{2}, \frac{9 + 1}{2} \right) = \left( \frac{8}{2}, \frac{10}{2} \right) = (4, 5)
\]
The center of the circle is at the point (4, 5). In the standard form of the circle's equation \((x-h)^2 + (y-k)^2 = r^2\), \(h\) represents the x-coordinate of the center, and \(k\) represents the y-coordinate of the center.
Thus, we have:
\[
h = 4
\]
So, the value of \(h\) is \(4\).
\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Substituting in the coordinates of the endpoints:
\[
\left( \frac{7 + 1}{2}, \frac{9 + 1}{2} \right) = \left( \frac{8}{2}, \frac{10}{2} \right) = (4, 5)
\]
The center of the circle is at the point (4, 5). In the standard form of the circle's equation \((x-h)^2 + (y-k)^2 = r^2\), \(h\) represents the x-coordinate of the center, and \(k\) represents the y-coordinate of the center.
Thus, we have:
\[
h = 4
\]
So, the value of \(h\) is \(4\).