Question 556808: WHAT IS THE CIRCUMCENTER OF TRIANGLE ABC WITH VERTICES A(-7,0), B(-3,8), AND C(-3,0)?
A. (-7,-3)
B. (-5,4)
C. (-4,3)
d. (-3,4)
THANK YOU
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! everything you ever wanted to know about circumcenter of a triangle is in the attached reference.
http://jwilson.coe.uga.edu/emat6680fa05/evans/assignment%204/assignment%204.htm
basically, the circumcenter of the triangle is the intersection of the 3 perpendicular bisectors of each side of the triangle.
if you draw a graph of your triangle, you will see that it has a horizontal side and a vertical side.
the midpoint of the horizontal side is (-5,0)
the midpoint of the vertical side is (-3,4)
if you extend a vertical line up from (-5,0) and you extend a horizontal line from (-3,4), those 2 lines will intersect aat (-5,4).
you could calculate the slope of the third line, but it should also intersect at that point since all 3 lines have to intersect at the same point.
my guess is that the circumcenter of the triangle is at the point (-5,4).
just to prove the point, i will find the equation of the perpendicular bisector of the line AC and show that it intersects the other 2 perpendicular bisectors at the point (-5,4).
the equation of the vertical line is x = -5
the equation of the horizontal line is y = 4
the equation of the perpendicular bisector of the third line needs to be calculated.
the third line is AB.
for the line AB:
(x1,y1) = (-7,0)
(x2,y2) = (-3,8)
the midpoint of line AB is equal to:
((x1+x2)/2,(y1+y1)/2)
this becomes:
((-10/2),(8/2)) which becomes:
(-5,4)
this point is also the intersection of the lines y = 4 and x = -5.
it appears the intersection of the perpendicular bisector of line AB with the line AB is on the line AB.
we should be able to see this on a graph.
the equation of the line AC is y = 0
the equation of the line BC is x = -3
the equation of the line AB can be generated as follows:
the slope of the line AB = (y2-y1)/(x2-x1) which becomes:
8/4 = 2
the y intercept of the line AB is the value of y when the value of x is equal to 0.
the slope intercept equation of the line AB is:
y = mx + b
m is the slope and b is the y intercept.
the slope is equal to 2 so the equation becomes:
y = 2x + b
we use any of the points on the lineto solve for b.
when the point is (-7,0), x = -7 and y = 0
the equation becomes:
0 = 2*(-7) + b which becomes:
0 = -14 + b which becomes:
b = 14
the equation of the line AB is therefore equal to y = 2x + 14
we can graph the equation of all 3 lines of:
y = 0
x = -3
y = 2x + 14
to get the graph or our triangle that is shown below:
now we will add the perpendicular bisector of the horizontal side of the triangle and the vertical side of the triangle as shown below:
you can see that the intersection of the perpendicular bisector is on the hypotenuse of the right triangle which is the midpoint of the hypotenuse which is the point (-5,4).
all 3 perpendicular bisectors intersect at the point (-5,4) so that point is the circumcenter of the triangle.
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