Question

Find the circumcenter and circumradius of the triangle whose vertices are (1,1),(2,-1) and (3,2).

Answer 1

Hint: Circumcenter is the center of a triangle's circumcircle. Circumradius is the radius of the circumscribed circle of that polygon.

Complete step-by-step answer:

We will be using the distance formula to evaluate the distance between the 2 points.
$D = \sqrt {{{\left( {{x_1} - {x_2}} \right)}^2} + {{\left( {{y_1} - {y_2}} \right)}^2}}$
Let A=(1,1), B=(2,-1) and C=(3,2).
Let S(x,y) be the circumcenter.
According to the question given, the condition is
$SA = SB$
${x^2} + {y^2} - 2x - 2y + 2 = {x^2} + {y^2} - 4x + 2y + 5$
$\Rightarrow 2x - 4y - 3 = 0$……(1)
$SA = SC$
${x^2} + {y^2} - 2x - 2y + 2 = {x^2} + {y^2} - 6x - 4y + 11$
$\Rightarrow 8x + 2y - 9 = 0$……(2)
From (1) and (2),
$\Rightarrow y = \dfrac{{ - 3}}{{18}},x = \dfrac{4}{3}$
Therefore, circumcenter is $S\left( {\dfrac{4}{3},\dfrac{{ - 3}}{{18}}} \right)$
To find the circumradius we are going to use the distance formula,
Circumradius,
$\sqrt {{{\left( {\dfrac{4}{3} - 1} \right)}^2} + {{\left( {\dfrac{{ - 3}}{{18}} - 1} \right)}^2}} = \sqrt {0.11 + 1.36} = 1.2$

Note: As we know circumcenter will be equal length from all vertices therefore we used that information to form equations to find the value of circumcenter and using the distance formula we calculate the circumradius.