   Chapter 10.2, Problem 44E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

Prove that if two lines (neither horizontal nor vertical) are perpendicular, then the product of their slopes is - 1 .(HINT: See Figure 10.19. You need to show and use the fact that ∆ A B C ~ ∆ E D C .)

To determine

To prove:

If two lines (neither horizontal nor vertical) are perpendicular, then the product of their slopes is -1.

Explanation

Consider the two perpendicular lines l1 and l2 which is neither vertical nor horizontal.

At the point of intersection C, draw a vertical CB and horizontal line CD.

From B, draw a horizontal line BA towards line l1, thus it makes the triangle ABC.

From D, draw a vertical line DE towards line l2, thus it makes the triangle CDE.

The C makes a right angle. Thus it is equal in both the triangle.

Also B and D makes a right angle because of the perpendicular bisector.

In ∆ABC and ∆CDE,

Two angles are same, obviously third angle is also equal.

A=E

By AAA similarity,

Corresponding sides of the triangle are proportional.

Thus, BCCD=ABDE

By interchanging the above ratio,

BCAB=CDDE

In this, we have to prove that the slopes are equal

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