Question 883375: How do you demonstrate that the perpendicular bisectors of the sides of any triangle with the points (-a,0) ; (b,c) ; (a,0) will intersect? I have tried to use the point-slope method to find the slope of each bisector, but come up with too many square roots of expressions subtracting too many variables.
English teacher trying to learn precalculus, Steve.
Thanks!
Answer by MathTherapy(10555)   (Show Source):
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How do you demonstrate that the perpendicular bisectors of the sides of any triangle with the points (-a,0) ; (b,c) ; (a,0) will intersect? I have tried to use the point-slope method to find the slope of each bisector, but come up with too many square roots of expressions subtracting too many variables.
English teacher trying to learn precalculus, Steve.
Thanks!
The perpendicular bisectors (PBs) of the sides of a triangle, with the PBs having points (- a, b), (b, c), and (a, 0) will INTERSECT INSIDE or OUTSIDE the triangle, at the CIRCUMCENTER, depending on whether the triangle is OBTUSE or ACUTE.
To prove that the PBs will intersect, let a be 5, b be 6, and c be 7. Therefore, the points (- a, b), (b, c), and (a, 0) become: (- 5, 6), (6, 7), and (5, 0), respectively.
Beginning with coordinate end-points, (- 5, 6) and (6, 7), find the coordinates of the MIDPOINT of this line. Then determine its slope. The equation to be determined will have a slope that is perpendicular to this slope. Using the coordinates of the midpoint, and the perpendicular slope, determine the equation of the perpendicular bisector of line with coordinate points (- 5, 6) and (6, 7).
Repeat the exact process with the side of the triangle having coordinate end-points: (- 5, 6) and (5, 0).
Again, repeat the exact process with the side of the triangle having coordinate end-points: (- 5, 0) and (6, 7).
Now, set two of the 3 equations equal to each other and determine the values of x and y.
Then, substitute these x and y values into the 3rd equation that WAS NOT used prior, to determine if the equation is TRUE. If the 3rd equation proves TRUE, based on the substituted x and y values, the THREE (3) equations INTERSECT at the x and y–coordinate point, just-found. Thus, the THREE (3) perpendicular bisectors of the sides of the triangle do INTERSECT, possibly INSIDE or OUTSIDE the triangle at a point called the CIRCUMCENTER.
If you’re confused by this explanation, let me know. I have created a graph on the coordinate plane to make the problem less complicated, through visualization.
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