Question

Find the center of the circle that can be circumscribed about triangle EFG with E(2, 2), F(2, -2), and G(6,-2).

Answer 1

Hello friend that would be the last option (3, 2)
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Answer 2

Answer:

The centre of circle is (4,0)          

Step-by-step explanation:

Given points are,

E(2, 2), F(2, -2), and G(6,-2).

The center of the circle is the circumcenter of the triangle, the point where the perpendicular bisectors intersect.

To find out the circumcenter we have to solve any two equations of bisectors of sides triangle.

$\text{Mid point of EF = }(\frac{2+2}{2},\frac{2-2}{2})= (2,0)$

$\text{Slope of EF = }\frac{y_2-y_1}{x_2-x_1}=\frac{-2-2}{2-2}=\frac{-4}{0}$

Slope of the bisector is negative reciprocal of the given slope.

So, the slope of the perpendicular bisector = 0

Equation of EF with slope 0 and the coordinates (2,0) is,

$(y - 0) = 0(x - 2)$

$y=0$   →    (1)

Similarly, for EG

$\text{Mid point of EG = }(\frac{2+6}{2},\frac{2-2}{2})= (4,0)$

$\text{Slope of EG = }\frac{y_2-y_1}{x_2-x_1}=\frac{-2-2}{6-2}=\frac{-4}{4}=-1$

So, the slope of the perpendicular bisector = 1

Equation of EG with slope 1 and the coordinates (4,0) is,

$(y - 0) =1(x - 4)$

$y=x-4$   →    (2)

Solving equation (1) and (2),

$0=x-4$

$x=4$

So the circumcenter is (4,0)

Option 2 is correct.