To find the equation of the circle given points \( A(1,1) \) and \( B(7,9) \), we can follow the steps outlined with the formulas provided.
Step 1: Measure of the Diameter
First, we will calculate the distance between points A and B, which represents the diameter of the circle. The formula for distance \( d \) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Plugging in the coordinates of points A and B:
- \( A(1,1) \) represents \( (x_1, y_1) \)
- \( B(7,9) \) represents \( (x_2, y_2) \)
Substituting the values:
\[ d = \sqrt{(7 - 1)^2 + (9 - 1)^2} \] \[ = \sqrt{(6)^2 + (8)^2} \] \[ = \sqrt{36 + 64} \] \[ = \sqrt{100} \] \[ = 10 \]
So, the measure of the diameter of the circle is 10.
Step 2: Measure of the Radius
The radius \( r \) is half of the diameter:
\[ r = \frac{d}{2} = \frac{10}{2} = 5 \]
So, the measure of the radius of the circle is 5.
Step 3: Center of the Circle (h,k)
Next, we find the midpoint of the diameter AB, which serves as the center of the circle. The midpoint formula is given by:
\[ (x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Calculating the midpoint for points A and B:
\[ (x, y) = \left( \frac{1 + 7}{2}, \frac{1 + 9}{2} \right) \] \[ = \left( \frac{8}{2}, \frac{10}{2} \right) \] \[ = (4, 5) \]
So, the center of the circle \( (h, k) \) is \( (4, 5) \).
Step 4: Equation of the Circle
The equation of a circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting the center \( (h, k) = (4, 5) \) and radius \( r = 5 \):
\[ (x - 4)^2 + (y - 5)^2 = 5^2 \] \[ (x - 4)^2 + (y - 5)^2 = 25 \]
Final Answers
- Measure of the diameter of the circle: 10
- Measure of the radius of the circle: 5
- Center of the circle (h,k): \( (4, 5) \)
- Write the entire equation of the circle: \( (x - 4)^2 + (y - 5)^2 = 25 \)