# A triangle has corners at #(2 , 1 )#, ( 5 , 6)#, and #( 8 , 5 )#. What are the endpoints and lengths of the triangle's perpendicular bisectors?

Endpoints at pairs of coordinates [

Thus, line 1 intersects with side CA, line 2 intersects with side BC, and line 3 intersects with side AB.

We require the equations of the three perpendicular lines as well as the lines in which the sides AB and CA lie.

identifying the interceptions on the AB and CA sides

Putting equations [1] and [c] together

Putting equations [2] and [c] together

Integrating the formulas [3] and [a]

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To find the endpoints of the perpendicular bisectors of the triangle, we first need to determine the midpoints of each side of the triangle. Then, we'll find the slopes of the sides and use those slopes to find the slopes of the perpendicular bisectors. Afterward, we'll use the midpoints and slopes to find the equations of the perpendicular bisectors. Finally, we'll solve the equations to find the endpoints of the bisectors.

Midpoint of the side joining (2, 1) and (5, 6): Midpoint = ((2 + 5)/2, (1 + 6)/2) = (3.5, 3.5)

Midpoint of the side joining (5, 6) and (8, 5): Midpoint = ((5 + 8)/2, (6 + 5)/2) = (6.5, 5.5)

Midpoint of the side joining (8, 5) and (2, 1): Midpoint = ((8 + 2)/2, (5 + 1)/2) = (5, 3)

Slope of the side joining (2, 1) and (5, 6): Slope = (6 - 1)/(5 - 2) = 5/3

Slope of the perpendicular bisector of this side = -1/(5/3) = -3/5

Equation of the perpendicular bisector passing through (3.5, 3.5) with slope -3/5: y - 3.5 = (-3/5)(x - 3.5)

Slope of the side joining (5, 6) and (8, 5): Slope = (5 - 6)/(8 - 5) = -1/3

Slope of the perpendicular bisector of this side = -1/(-1/3) = 3

Equation of the perpendicular bisector passing through (6.5, 5.5) with slope 3: y - 5.5 = 3(x - 6.5)

Slope of the side joining (8, 5) and (2, 1): Slope = (1 - 5)/(2 - 8) = 4/-6 = -2/3

Slope of the perpendicular bisector of this side = -1/(-2/3) = 3/2

Equation of the perpendicular bisector passing through (5, 3) with slope 3/2: y - 3 = (3/2)(x - 5)

Solving these equations will give us the endpoints of the perpendicular bisectors.

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