PROOF 22

Prove.

Perpendicular lines have slopes whose product is −1

On the coordinate plane.

What you may use

A1In a right triangle, a² + b² = c²
Pythagorean theorem (cited here, not proved)
A2horizontal segment ⊥ vertical segment
The coordinate axes are perpendicular
A3horizontal: |x₂−x₁|, vertical: |y₂−y₁|
Length of a segment parallel to an axis
A4|t|² = t²
Square of the absolute value
A5a(b + c) = ab + ac
Distributive law
A6ab = ba
Commutativity of multiplication
A7a × 1 = a
Multiplicative identity
A8a(−b) = −(ab)
Sign rule 1
A9(−a)(−b) = ab
Sign rule 2
A10(a+b)+c = a+(b+c)
Associativity of addition
A11a + b = b + a
Commutativity of addition
A12a + b = a + c ⟹ b = c
Cancellation law for addition
A13ca = cb, c ≠ 0 ⟹ a = b
Cancellation law for multiplication
A14the −a with a + (−a) = 0 is unique
Additive inverse
A15a + 0 = a, 0² = 0, 0 ≠ 2
Properties of 0
Ea = b ⟹ same operation on both sides
Properties of equality

Nothing outside this list may be cited.

Definition D1

The line through the origin with slope m is y = mx, and it passes through the point (1, m).

Proof.