A triangle has corners at (7,7), (1,3), and (6,5). What is the area of the triangle's circumscribed circle?

1 Answer
Shiva Prakash M V
Feb 7, 2018

Area of the circum circle is 158.503

Explanation:

Midpoint of corners (7,7)and(1,3) is given by
(7+32,7+12)
=(5,4)
Slope of the line between the corners (7,7)and(1,3) is given by
3−71−7
=−4−6
=23
Slope of the perpendicular bisector for the line between the corners (7,7)and(1,3) is given by
=−123
=−32
We have the equation of the perpendicular bisector for the line between the corners (7,7)and(1,3) given by
Point (5,4)
Slope −32
point slope form is
y−4x−5=−32

Midpoint of corners (1,3)and(6,5) is given by
(1+62,3+52)
=(3.5,4)
Slope of the line between the corners (1,3)and(6,5) is given by
5−36−1
=25
=25
Slope of the perpendicular bisector for the line between the corners (1,3)and(6,5) is given by
=−125
=−52
We have the equation of the perpendicular bisector for the line between the corners (1,3)and(6,5) given by
Point (3.5,4)
Slope −52
point slope form is
y−4x−3.5=−52

Solving for the center of the circumcircle
y−4x−5=−32
2(y−4)=−3(x−5)
2y−8=−3x+15
2y+3x−15−8=0
3x+2y−23=0

y−4x−3.5=−52
2(y−4)=−5(x−3.5)
2(y−4)+5(x−3.5)=0
2y−8+5x−17.5=0
5x+2y−24.5=0

We have,
3x+2y−23=0
5x+2y−24.5=0
Eliminating y
3x−5x−23+24.5=0
−2x+1.5=0
2x=1.5
x=0.75
Substituting
3(0.75)+2y−23=0
2.25+2y−23=0
Simplifying
2y=23−2.25
2y=20.75
y=10.375
Center for the circum circle has the coordinates
(0.75,10.375)
Radius r=distance from center to vertex
(0.75,10.375)−−−−(7,7)
Area of the circum circle is
A=πr2
π=3.141
r2=(7−0.75)2+(7−10.375)2
r2=(6.25)2+(−3.375)2
r2=50.453
Hence, Area is
A=3.141⋅50.453
Area of the circum circle is 158.503

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