   Chapter 1.4, Problem 90E

Chapter
Section
Textbook Problem

Prove that if a line L1 with slope m1 is perpendicular to a line L2 with slope m2, then m1m2 = −1.Hint: Refer to the following figure. Show that m1 = b and m2 = 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 L1 with slope m1 and L2 with slope m2 is perpendicular to each other then m1m2=1 .

Explanation

Proof:

The given figure is shown below in Figure 1,

Figure (1)

The points A(1,b) and O(0,0) are on the line L1 .

The slope of a line when two passing points are (x1,y1) and (x2,y2) is given by,

m=y2y1x2x1 (1)

Substitute m1 for m, 1 for x1 , b for y1 , 0 for x2 , and 0 for y2 in equation (1).

The slope of the line L1 is calculated as follows.

m1=0b01=b1=b

Thus, the slope of line L1 is b.

The points B(1,c) and O(0,0) are on the line L2 .

Substitute m2 for m, 1 for x1 , c for y1 , 0 for x2 , and 0 for y2 in equation (1).

The slope of the line L2 is calculated as follows.

m2=0c01=c1=c

Thus, the slope of line L2 is c.

Use distance formula to find the distance between A(1,b) , O(0,0) and B(1,c) , O(0,0) .

Distance between two points (x1,y1) and (x2,y2) is given by,

d=(y2y1)2+(x2x1)2 (2)

Substitute 1 for x1 , b for y1 , 0 for x2 , and 0 for y2 in equation (2).

The distance between A and O is calculated as follows.

d=(0b)2+(01)2=b2+12=1+b2

The length of side AO is 1+b2

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