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

• plot the 3 points(optional)
• use the distance formula to calculate the side length of each side of the triangle.
• If any 2 sides have equal side lengths, then the triangle is isosceles.

Practice 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.

Step 1) Plot Points

Calculate all 3 distances

Step 2) calculate the distances

The distances

Step 4) Draw your conclusion:

Triangle BCD is isosceles because 2 of the sides have equal side length (ie are congruent)

Practice 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.

Step 1) Plot Points

Calculate all 3 distances

Step 2) Show 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)

Step 4) Draw your conclusion

Triangle ABC is isosceles because 2 of the sides have equal side length (ie are congruent)