Question

Use the following formulas to help you find the equation of a circle give two points that are on the diameter of the circle.
distance: d=(x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√
midpoint: (x,y)=(x1+x22,y1+y22)
circle: (x−h)2+(y−k)2=r2(5 points)
Write the equation for a circle with a diameter AB¯¯¯¯¯¯¯¯ given coordinates A(1,1) and B(7,9).

Measure of the diameter of the circle:

Measure of the radius of the circle:

Center of the circle (h,k): (
,
)

Write the entire equation of the circle:

Answer 1

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: \( (x - 4)^2 + (y - 5)^2 = 25 \)