Q 128 - Precalculus Enhanced with Graphing Utilities

Question

Prove that if two nonvertical lines have slopes whose product is -1 then the lines are perpendicular.

Step-by-Step Solution

Answer

OA and OB are perpendicular

Step 1. Given information

Prove that if two nonvertical lines have slopes whose product is -1 then the lines are perpendicular.

Step 2. According to the figure 65 of the text book

The slope of line 1 is $m_1$ and the slope of the line 2 is $m_2$

In triangle OAB

$$\begin{gathered} \left[d\right(O,A){]}^2=(1-0{)}^2+{\left(m_2-0\right)}^2=1+m_2^2 \\ \left[d\right(O,B){]}^2=(1-0{)}^2+{\left(m_1-0\right)}^2=1+m_1^2 \\ \left[d\right(A,B){]}^2=(1-1{)}^2+{\left(m_2-m_1\right)}^2=m_2^2+m_1^2-2m_1{m}_2 \end{gathered}$$

Now $m_1{m}_2=-1$

Therefore, 

$\left[d\right(A,B){]}^2=(1-1{)}^2+{\left(m_2-m_1\right)}^2=m_2^2+m_1^2+2$

so we can see

$\left[d\right(O,A){]}^2+[d(O,B){]}^2=m_2^2+m_1^2+2=\left[d\right(A,B){]}^2$

Therefore OA and OB are perpendicular.