Answer:
(1, 6.5)
Explanation:
The point equidistant from the three vertices of a triangle is called the circumcenter of the triangle.
It is located at the intersection of the perpendicular bisectors of the three sides of the triangle.
Since all three perpendicular bisectors of the sides of a triangle intersect at one point, all you need to do is to find the equations of two of the perpendicular bisectors and solve them as a system of equations. The solution is the circumcenter of the triangle.
Let's find the equation of the perpendicular bisector of side EF.
E(-5, 9), F(1, 0)
midpoint = ((-5 + 1)/2, (9 + 0)/2) = (-2, 4.5)
slope of segment EF: m = (9 - 0)/(-5 - 1) = 9/(-6) = -3/2
slope of perpendicular bisector: 2/3
equation of perpendicular bisector:
y = mx + b
4.5 = (2/3)(-2) + b
13.5 = -4 + 3b
b = 35/6
y = (2/3)x + 35/6 Eq. 1
Now let's find the equation of the perpendicular bisector of side FG.
F(1, 0), G(7, 4)
midpoint = ((1 + 7)/2, (0 + 4)/2) = (4, 2)
slope of segment FG: m = (4 - 0)/(7 - 1) = 4/6 = 2/3
slope of perpendicular bisector: -3/2
equation of perpendicular bisector:
y = mx + b
2 = (-3/2)(4) + b
2 = -6 + b
b = 8
y = (-3/2)x + 8 Eq. 2
Equations 1 and 2 are the equations of two perpendicular bisectors of two sides of the triangle. Now we solve them as a system of simultaneous equations to find the point of intersection.
y = (2/3)x + 35/6
y = (-3/2)x + 8
Since both equations are solved for y, equate the right sides.
(2/3)x + 35/6 = (-3/2)x + 8
Multiply both sides by 6, the LCD, to get rid of all denominators.
6 × (2/3)x + 6 × 35/6 = 6 × (-3/2)x + 6 × 8
4x + 35 = -9x + 48
13x = 13
x = 1
y = (-3/2)x + 8
y = (-3/2)(1) + 8
y = 6.5
Answer: (1, 6.5)