Name: _____________________________________
Geometry
Summary of Methods used for Coordinate Geometry Proofs
To prove that a triangle is an isosceles or equilateral triangle, use:
 the distance formula to show that two or three sides have equal lengths/congruent
To prove that a triangle is a right triangle, use:
 the distance formula to verify the Pythagorean Theorem (or)
 slopes to show that two sides of the triangle are perpendicular
To prove that a quadrilateral is a parallelogram, use:
 the distance formula to show that both pairs of opposite sides are congruent (or)
 the midpoint formula to show that the diagonals bisect each other (or)
 slopes to show that both pairs of opposite sides are parallel (or)
 slopes to show that one pair of opposite sides are parallel and use the distance formula to show
these same sides are congruent
To prove that a parallelogram is a rectangle, use:
 the distance formula to show that the diagonals of the parallelogram are congruent (or)
 slopes to show that two consecutive sides of the parallelogram are perpendicular
To prove that a parallelogram is a rhombus, use:
 the distance formula to show that two consecutive sides of the parallelogram are congruent (or)
 slopes to show that the diagonals of the parallelogram are perpendicular
To prove that a parallelogram is a square, use:
 one method to prove that it’s a rectangle and one method to prove that it’s a rhombus
To prove that a quadrilateral is a trapezoid, use:
 slope to show that one, and only one, pair of opposite sides are parallel
To prove that a trapezoid is isosceles, use:
 distance formula to show the non-parallel sides of the trapezoid are congruent
Summary of Techniques and Formulas
1. To prove that line segments are congruent, show that the
Distance:
segments are equal in length by using the distance formula.
2. To prove that segments bisect each other, show that the
same ordered pair represents the midpoint of each line
segment.
3. To prove that lines intersect at a point, write an equation of
each line and solve the system of equations.
Midpoint:
(xm, ym) =
4. To prove that lines are parallel, show that the slopes of the
lines are equal.
5. To prove that lines are perpendicular, show that their slopes
are negative reciprocals, or that the product of the slopes is -1.
Slopes: m1 = m2
Line: y = mx + b
Slopes:
1
m1 = - 𝑚2
m = slope, b = yintercept
or
m1  m2 = -1