Chapter 1.4, Problem 90E

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10th Edition

Soo T. Tan

ISBN: 9781285464640

Textbook Problem

Prove that if a line *L*_{1} with slope *m*_{1} is perpendicular to a line *L*_{2} with slope *m*_{2}, then *m*_{1}*m*_{2} = −1.

**Hint:** Refer to the following figure. Show that *m*_{1} = *b* and *m*_{2} = *c*. Next, apply the Pythagorean Theorem to triangles *OAC*, *OCB*, and *OBA* to show that 1 = −*bc*.

To determine

**To show:** If two lines namely

Explanation

**Proof:**

The given figure is shown below in Figure 1,

Figure (1)

The points

The slope of a line when two passing points are

Substitute
*m*, 1 for
*b* for

The slope of the line

Thus, the slope of line
*b*.

The points

Substitute
*m*, 1 for
*c* for

The slope of the line

Thus, the slope of line
*c*.

Use distance formula to find the distance between

Distance between two points

Substitute 1 for
*b* for

The distance between *A* and *O* is calculated as follows.

The length of side *AO* is

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P-56ESect-1.4 P-57ESect-1.4 P-58ESect-1.4 P-59ESect-1.4 P-60ESect-1.4 P-61ESect-1.4 P-62ESect-1.4 P-63ESect-1.4 P-64ESect-1.4 P-65ESect-1.4 P-66ESect-1.4 P-67ESect-1.4 P-68ESect-1.4 P-69ESect-1.4 P-70ESect-1.4 P-71ESect-1.4 P-72ESect-1.4 P-73ESect-1.4 P-74ESect-1.4 P-75ESect-1.4 P-76ESect-1.4 P-77ESect-1.4 P-78ESect-1.4 P-79ESect-1.4 P-80ESect-1.4 P-81ESect-1.4 P-82ESect-1.4 P-83ESect-1.4 P-84ESect-1.4 P-85ESect-1.4 P-86ESect-1.4 P-87ESect-1.4 P-88ESect-1.4 P-89ESect-1.4 P-90ECh-1 P-1CRQCh-1 P-2CRQCh-1 P-3CRQCh-1 P-4CRQCh-1 P-5CRQCh-1 P-6CRQCh-1 P-7CRQCh-1 P-8CRQCh-1 P-1RECh-1 P-2RECh-1 P-3RECh-1 P-4RECh-1 P-5RECh-1 P-6RECh-1 P-7RECh-1 P-8RECh-1 P-9RECh-1 P-10RECh-1 P-11RECh-1 P-12RECh-1 P-13RECh-1 P-14RECh-1 P-15RECh-1 P-16RECh-1 P-17RECh-1 P-18RECh-1 P-19RECh-1 P-20RECh-1 P-21RECh-1 P-22RECh-1 P-23RECh-1 P-24RECh-1 P-25RECh-1 P-26RECh-1 P-27RECh-1 P-28RECh-1 P-29RECh-1 P-30RECh-1 P-31RECh-1 P-32RECh-1 P-33RECh-1 P-34RECh-1 P-35RECh-1 P-36RECh-1 P-37RECh-1 P-38RECh-1 P-39RECh-1 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