Find the equation of the circle that has diameter endpoints of ( 4, 8 ) and (10, 16 )

Step-by-step explanation:

The circle has diameter endpoints of ( 4, 8 ) and (10, 16 ).

The midpoint of the diameter is the center

The radius is the distance from the center to any point on the circumference.

The equation of the circle with center (h,k) and radius r is given by:

We plug in the values to get:

The equation of the circle is (x - 7)² + (y - 12)² = 25

Step-by-step explanation:

* Lets revise the equation of the circle

- The equation of the circle which has center (h , k) is:

 (x - h)² + (y - k)² = r²

- The endpoints of a diameter re (4 , 8) and (10 , 16)

- The center of the circle is the mid-point of the diameter

* Lets revise the rule of the mid point to get the center of the circle

 and revise the rule of the distance to get the length of the diameter

- The mid point (x , y) of segment has endpoints (x1 , y1) and (x2 , y2) is

 x = (x1 + x2)/2  and y = (y1 + y2)/2

- The distance between the two point (x1 , y1) and (x2 , y2) is:

  d = √[(x2 - x1)² + (y2 - y1)²]

* Now lets find the center of the circle

∵ (h , k) is the mid-point of the diameter

∵ (4 , 8) is (x1 , y1) and (10 , 16) is (x2 , y2)

∴ h = (4 + 10)/2 = 14/2 = 7

∴ k = (8 + 16)/2 = 24/2 = 12

∴ The center of the circle is (7 , 12)

∵ The length of the diameter = √[(10 - 4)² + (16 - 8)²

∴ d = √[6² + 8²] = √[36 + 64] = √100 = 10

∵ The radius = 1/2 the diameter

∴ r = 1/2 (10) = 5

* Lets write the equation of the circle

∵ The equation of the circle is (x - h)² + (y - k)² = r²

∵ h = 7 , k = 12 , r = 5

∴ The equation of the circle is (x - 7)² + (y - 12)² = 5²

∴ The equation of the circle is (x - 7)² + (y - 12)² = 25

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step-by-step explanation: