Answer:
The equation of the circle is (x - 4)² + (y - 9)² = 4²
Step-by-step explanation:
The center radius form of the equation of the circle is (x - h)² + (y - k)² = r²
Therefore, given the coordinates of the vertices of the inscribed triangle, we have points on the circumference includes;
X(4, 5), Y(4, 13), Z(8, 9)
Hence, we have;
(4 - h)² + (5 - k)² = r² = h²- 8·h - 10·k + k² + 41..........(1)
(4 - h)² + (13 - k)² = r² = h² - 8·h + k² - 26·k +185.....(2)
(8 - h)² + (9 - k)² = r² = h² - 16·h -18·k + k² +145.........(3)
Subtracting equation (1) from (2), we have;
-(16·k -144) = 0
∴ 16 k = 144
k = 9
Subtracting equation (1) from (3), we have;
-(8·h + 8·k - 104) = 0
Which gives;
8·h + 8·k - 104 = 0, where k = 9 gives
8·h + 8×9 - 104 = 0
8·h - 32 = 0
h = 32/8 = 4
Plugging in the values of h and k in equation (3), we have;
(8 - 4)² + (9 - 9)² = r²
4² = r²
∴ r = 4
The equation of the circle is (x - 4)² + (y - 9)² = 4².