Ok, so I have this question from my math book.
The vertices A,B,C of a triangle are (3,5),(2,6) and (-4,-2) respectively. Find the coordinates of the circum-centre and also the radius of the circum-circle of the triangle.
How can we solve this? Can we use the distance formula? Answer: The circum-radius was found to be R=5.The coordinates of circum-centre were found to be (-1,2). A diagram would be appreciated. Thank you!
On
HINT...You could find the equations of the perpendicular bisectors of two of the sides and where they meet will be the circumcentre. Then use the distance formula to work out the circumradius.
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So you can find the linear equations of the segment bisectors of each edge (really you only need two) which are $$ y=-x+1 $$ $$ y=x+3 $$ $$ y=-\frac{3x}{4} + \frac{5}{4} $$ Interestingly, these all coincide at the midpoint of segment $\overline{BC}$ at $(-1,2)$ as shown below (graph available here: https://www.desmos.com/calculator/gqwj7pn6ud)
The radius of the circle is the distance from there to any one of the vertices of the triangle which gives you $r=5$ so the equation of the circle is given as
$$
(x+1)^2+(y-2)^2 = 5^2
$$
Hint:
You want a circumference that passe thorough the three given points.
The general equation of a circumference is $x^2+y^2+ax+by+c=0$.
Substitute the coordinates of the three points and you have a linear system in the three unknowns $a,b,c$.
Solve this system and you have the equation of the circumference from wich you can find the center and the radius.