Answer:
(3,2) is the center of the circle that can be circumscribed about triangle ABC with A(0, 0), B(6, 0), C(6, 4).
Step-by-step explanation:
The standard equation of a circle with center (h, k) and radius "r" is given by:
(x - h)²+ (y - k)² = r²
By using the 3 unknown points on the circle, we can have three equations that have to be solved simultaneously.
So,
Using point A(0, 0) in standard equation of a circle:
(x - h)²+ (y - k)² = r²
(0 - h)²+ (0 - k)² = r²
h² + k² = r² Equation ( 1 )
Using point B(6, 0) in standard equation of a circle:
(x - h)²+ (y - k)² = r²
(6 - h)²+ (0 - k)² = r²
36 - 12h + h² + k² = r² Equation ( 2 )
Using point B(6, 4) in standard equation of a circle:
(x - h)²+ (y - k)² = r²
(6 - h)²+ (4 - k)² = r²
36 - 12h + h² + 16 - 8K + k² = r²
52 - 12h + h² - 8K + k² = r² Equation ( 3 )
Subtracting Equation ( 1 ) from Equation ( 2 )
36 - 12h + h² + k² = r²
- h² ± k² = - r²
···············································
36 - 12h = 0 Equation ( A )
···············································
Subtracting Equation ( 1 ) from Equation ( 3 )
52 - 12h + h² - 8k + k² = r²
- h² ± k²= - r²
···············································
52 - 12h - 8k = 0 Equation ( B )
···············································
By Equation (A)
36 - 12h = 0
12h = 36
h = 3
Putting the equation h = 3 in Equation (B)
52 - 12h - 8k = 0
52 - 12(3) - 8k = 0
52 - 36 = 8k
16 = 8k
k = 2
So, the center of circle = (h, k) = (3, 2)
Keywords: circle, center of circle, triangle
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