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# Analytic Geometry

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Determine the point of intersection of right bisectors in a triangle ∆𝐴𝐵𝐶 with vertices A (-3, 5), B (1, 1) and 𝐶(−7, −3). Find the distance from the point of intersection to each vertex of the triangle.

Apr 15, 2021

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Find  the  midpoint  of AB   =  (-1, 3)

Find  the slope of AB  =   4/ -4  =  -1

A perpendicular line  to AB  through the midpoint of AB  has  the equation

y = 1 (x - -1)  + 3

y = x + 4       (1)

Find  the  midpoint of BC  = ( -3,  -1)

Slope  of BC   =  ( -4/ -8)  = 1/2

A perpedicular line to BC   through  the  midpoint of BC has the equation

y = (2) ( x - - 3)  -1

y = (-2)( x + 3)  - 1

y = -2x -7     (2)

Find  the x  intersection of (1)  and (2)

x + 4  =  -2x - 7

3x = -11

x  = -11/3

And  y  = (-11/3) + 4   =   1/3

So.....the point of intersection  ( the  circumcenter)  =  ( -11/3, 1/3)

The distance  from this point to each  vertex  is  the  same  =

sqrt [ ( -11/3 - 1)^2  + (1/3 - 1)^2  ]  =   sqrt  [  (14/3)^2  + ( 2/3)^2 ]  = sqrt  [  14^2 + 2^2]  / 3   =

sqrt [ 200] / 3   =   10sqrt (2)  /  3

Apr 15, 2021