   Chapter 10.2, Problem 38E 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

Three vertices of rectangle A B C D are A - 5 ,   1 , B - 2 ,   - 3 , and C 6 ,   y . Find the value of y and also the fourth vertex.

To determine

To find:

The value of y and also the fourth vertex for the given three vertices A-5, 1, B-2, -3, and C6, y of a rectangle ABCD.

Explanation

In a rectangle, two adjacent sides are perpendicular to each other.

By theorem,

If two lines are perpendicular, then the product of their slopes is -1 or one slope is negative reciprocal of the other slope.

(i.e) If l1l2, then m1.m2=-1 or m2=-1m1

Thus we need to check the slopes of any two sides with right angle at B.

The slope of the line that contains the points x1,y1 and x2,y2 is given by

m=y2-y1x2-x1 for x2x1

Let mAB- and mBC- are the slopes of the line AB- and BC- respectively.

The given points are A-5, 1, and B-2, -3.

Using the slope formula and choosing x1=-5, x2=-2, y1=1, and y2=-3.

mAB-=-3-1-2--5

mAB-=-43

The given points are B-2, -3 and C6, y.

Using the slope formula and choosing x1=-2, x2=6, y1=-3, and y2=y.

mBC-=y--36--2

mBC-=y+38

mAB-. mBC-=-1

-43y+38=-1

On multiplying,

-4y-1224=-1

On cross multiplying,

-4y-12=-24

Add 12 to both sides,

-4y-12+12=-24+12

-4y=-12

Divide both sides by -4,

-4y-4=-12-4

y=3

Thus the value of y is 3

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