Write an equation of a circle with diameter AB.A(0,0),B(-6,8)

To find the equation of a circle with diameter AB, we need to find the center coordinates and the radius.

Step 1: Find the center coordinates:
The center of the circle is the midpoint between the two endpoints of the diameter. The midpoint formula is given by:

Midpoint (x, y) = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

Using the coordinates of A(0,0) and B(-6,8), we can find the midpoint:

x = (0 + (-6)) / 2 = -3
y = (0 + 8) / 2 = 4

So the center of the circle is (-3, 4).

Step 2: Find the radius:
The radius of the circle is half the length of the diameter. The distance formula is used to find the length of the diameter:

Distance = √((x₂ - x₁)² + (y₂ - y₁)²)

Using the coordinates of A(0,0) and B(-6,8), we can find the distance:

Distance = √((-6 - 0)² + (8 - 0)²)
= √((-6)² + 8²)
= √(36 + 64)
= √100
= 10

So the radius of the circle is 10 units.

Step 3: Write the equation:
The equation of a circle with center (h, k) and radius r is:

(x - h)² + (y - k)² = r²

Using the values we found in Step 1 and Step 2, the equation of the circle is:

(x - (-3))² + (y - 4)² = 10²

Simplifying, we have:

(x + 3)² + (y - 4)² = 100

Therefore, the equation of the circle with diameter AB (A(0,0), B(-6,8)) is (x + 3)² + (y - 4)² = 100.

To find the equation of a circle with diameter AB, we first need to find the center of the circle. The center of a circle is found by taking the midpoint of the diameter. In this case, the coordinates of A are (0,0) and the coordinates of B are (-6,8).

To find the midpoint, we use the midpoint formula:
Midpoint(x,y) = [ (x1 + x2)/2 , (y1 + y2)/2 ]

Using the coordinates of A and B, we substitute them into the midpoint formula:
Midpoint(x,y) = [ (0 + (-6))/2 , (0 + 8)/2 ]
Midpoint(x,y) = [-6/2 , 8/2 ]
Midpoint(x,y) = [-3, 4 ]

So, the center of the circle is (-3,4).

Now that we have the center, we can write the equation of the circle. The general equation of a circle is:
(x - h)² + (y - k)² = r²

Where (h,k) is the center of the circle, and r is the radius. Since the diameter is given, we can find the radius by finding the distance between the center and any point on the circle (in this case, A or B).

Using the distance formula:
Distance = sqrt( (x2 - x1)² + (y2 - y1)² )

Let's find the distance between the center (-3,4) and point A (0,0):
Distance = sqrt( (0 - -3)² + (0 - 4)² )
Distance = sqrt( (0 + 3)² + (-4)² )
Distance = sqrt( 3² + 16 )
Distance = sqrt( 9 + 16 )
Distance = sqrt( 25 )
Distance = 5

So, the radius of the circle is 5.

Now we can substitute the values for the center (-3,4) and the radius 5 into the equation of a circle:
(x - (-3))² + (y - 4)² = 5²
(x + 3)² + (y - 4)² = 25

Therefore, the equation of the circle with diameter AB, where A(0,0) and B(-6,8), is (x + 3)² + (y - 4)² = 25.

as you know from your 3-4-5 triangles, the diameter length is 10. So, the midpoint will be at (-3,4) and the circle is

(x+3)^2 + (y-4)^2 = 25