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Prove Triangle Is Isosceles using Coordinate Geometry

An isosceles triangle has 2 congruent sides and two congruent angles. The easiest way to prove that a triangle is isosceles using coordinate geometry is to use the sides.

Steps to Coordinate Proof

Given the coordinates of the triangle's vertices, to prove that a triangle is isosceles
Problem 1
The coordinates of triangle BCD are B(8,2), C(11,13) and D(2,6). Using coordinate geometry, prove that triangle BCD is an isosceles triangle.
Plot points
Calculate all 3 distances
Draw your conclusion
Triangle BCD is isosceles because 2 of the sides have equal side length $$ \sqrt{130} $$ (ie are congruent)
Problem 2
Triangle ABC has coordinate A(-2,3) , B (-5,-4) and C (2,-1). Using coordinate geometry, prove that triangle BCD is an isosceles triangle.
Plot points
isosceles triangle Coordinate Geometry
Calculate all 3 distances
The distances
If you calculated side lengths AB and BC first, you could stop -- the triangle must be isosceles at this point since you found 2 sides that are congruent. (Granted, the triangle still could be equilateral)
Draw your conclusion
Triangle ABC is isosceles because 2 of the sides have equal side length $$ \sqrt{58} $$ (ie are congruent)
Back to Geometry Next to Euclidean Proof