A triangle has corners at #(9 ,5 )#, #(2 ,3 )#, and #(7 ,6 )#. What is the area of the triangle's circumscribed circle?
2 Answers
Explanation:
The standard Cartesian form for the equation of a circle is:
where
We can use the points,
Expand the squares:
Subtract equation [3.1] from equation [2.1]:
Combine like terms:
Subtract equation [3.1] from equation [4.1]:
Combine like terms:
Multiply both sides of equation [5] by
#72 - 10h -6k - 3/2(93 - 14h - 4k) = 0
Distribute the
#72 - 10h -6k - 279/2 + 21h + 6k = 0
Combine like terms:
Use equation [6] to find the value of k:
Use equation [2] to find the value of
Because the area of a circle is
Circum center (6.1364, 1.7727)
Area of circumcircle 58.5065
Explanation:
Slope of AB m1 = (3-5)/(2-9)=2/7.
Slope of perpendicular at mid point of AB = -1/m1 = -7/2
Midpoint of AB = (9+2)/2, (3+5)/2 = 11/2, 4
Eqn of perpendicular bisector of AB is
Slope of BC m2 = (6-3)/(7-2) = 3/5.
Slope of perpendicular at mid point of AB = -1/m1 = -5/3.
Midpoint of BC = (7+2)/2, (6+3)/2 = 9/2, 9/2
Eqn of perpendicular bisector of BC is
Solving Eqns (1), (2)
Circum center (6.1364, 1.7727)
Area of circumcircle