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Chapter 1: Q 128 (page 43)

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

Short Answer

Expert verified

OA and OB are perpendicular

Step by step solution

01

Step 1. Given information

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

02

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

$[d (O, A)]^{2} = (1 - 0)^{2} + {(m_{2} - 0)}^{2} = 1 + m_{2}^{2} [d (O, B)]^{2} = (1 - 0)^{2} + {(m_{1} - 0)}^{2} = 1 + m_{1}^{2} [d (A, B)]^{2} = (1 - 1)^{2} + {(m_{2} - m_{1})}^{2} = m_{2}^{2} + m_{1}^{2} - 2 m_{1} m_{2}$

Now $m_{1} m_{2} = - 1$

Therefore,

$[d (A, B)]^{2} = (1 - 1)^{2} + {(m_{2} - m_{1})}^{2} = m_{2}^{2} + m_{1}^{2} + 2$

so we can see

role="math" localid="1649411957539" $[d (O, A)]^{2} + [d (O, B)]^{2} = m_{2}^{2} + m_{1}^{2} + 2 = [d (A, B)]^{2}$

Therefore OA and OB are perpendicular.

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Most popular questions from this chapter

What can you say about two lines with the same $x$ -intercept and the same $y$ -intercept Assume that the $x$ -intercept is not 0.

Plot the points $(0, 3), (1, 3), (- 2, 3), (5, 3),$ and $(- 4, 3)$ . Describe the set of all points of the form $(x, 3)$ , where x is a real number.

Which of the following equations might have the graph shown? (More than one answer is possible.)

$(a) 2 x + 3 y = 6 (b) - 2 x + 3 y = 6 (c) 3 x - 4 y = - 12 (d) x - y = 1 (e) x - y = - 1 (f) y = 3 x - 5 (g) y = 2 x + 3 (h) y = - 3 x + 3$

Solve:

$x^{2} - 4 x - 12 = 0$

If $- 3$ and 5 are the coordinates of two points on the real number line, the distance between these points is _________.

See all solutions

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