Math 1 · Unit 4 · Lesson 4 · part 2 of 5
Slope can prove a shape
Slope can prove things about shapes.
Slope can prove a shape
Slope can prove things about shapes. Give the corners of a shape as points on the coordinate plane, and the slopes of its sides tell you how the sides are related, exactly as in Lesson 3. Two sides with equal slopes are parallel. Two sides whose slopes multiply to −1 meet at a right angle. That is enough to test claims like these:
Right triangleA triangle with one right angle. Test: do two of its sides have slopes that are negative reciprocals? If yes, the corner where they meet is the right angle.
RectangleFour right angles, and opposite sides parallel. Test: are the slopes of opposite sides equal, and do neighboring sides have slopes that multiply to −1?
ParallelogramOpposite sides parallel. Test: equal slopes for each pair of opposite sides. (No right angles needed.)
Proving with algebra. Drawing a shape and saying it looks like a rectangle is a guess. Computing four slopes and showing the opposite ones are equal and the neighbors multiply to −1 is a proof: every claim rests on a number you found. That is what it means to prove a geometric fact with algebra, using only slope.