Slopes of Perpendicular Lines Theorem

Since the theorem is a biconditional statement, the proof consists of two parts.

1. If two non-vertical lines are perpendicular, then the product of their slopes is - 1.
2. If the product of the slopes of two non-vertical lines is - 1, then the lines are perpendicular.

Part 1

Let l_1 and l_2 be two perpendicular lines. Therefore, they intersect at one point. For simplicity, the lines will be translated so that the point of intersection is the origin.

Let m_1 and m_2 be the slopes of the lines l_1 and l_2, respectively. Next, consider the vertical line x=1. This line intersects both l_1 and l_2.

Since l_1 and l_2 are assumed to be perpendicular, △ AOC is a right triangle. Using the Distance Formula, the lengths of the sides of this triangle can be found.

Side	Points	Distance Formula sqrt((x_2-x_1)^2+(y_2-y_1)^2)	Length
AO	A( 1, m_1) & O( 0, 0)	sqrt(( 0- 1)^2+( 0- m_1)^2)	sqrt(1+m_1^2)
CO	C( 1, m_2) & O( 0, 0)	sqrt(( 0- 0)^2+( 0- m_2)^2)	sqrt(1+m_2^2)
CA	C( 1, m_2) & A( 1, m_1)	sqrt(( 1- 1)^2+( m_1- m_2)^2)	m_1-m_2

Since △ AOC is a right triangle, its side lengths satisfy the Pythagorean Equation. AO^2+ CO^2 = CA^2 The next step is to substitute the lengths shown in the table.

It has been proven that if two lines are perpendicular, then the product of their slopes is - 1.

Part 2

Here it is assumed that the slopes of two lines l_1 and l_2 are opposite reciprocals. m_1* m_2 =- 1 Consider the steps taken in Part 1. This time, it should be found that △ AOC is a right triangle.

If the lengths of the sides of △ AOC satisfy the Pythagorean Theorem, then the triangle is a right triangle. AO^2+ CO^2 ? = CA^2 The side lengths, which were previously found in Part 1, can be substituted into the above equation.

Since a true statement was obtained, △ AOC is a right triangle. Therefore, l_1 and l_2 are perpendicular lines. This completes the second part.

The biconditional statement has been proven.