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Construct a Perpendicular Segment

Example

Example

2

2

COORDINATE GEOMETRY

Line contains points ( 6,

9) and (0,

1).

Construct a line perpendicular to line through P( 7,

2) not on .

Then find the distance from P to .

Graph line

and point P. Place the compass

y

1

1

4

point at point P. Make the setting wide enough

so that when an arc is drawn, it intersects in two

O

x

P

places. Label these points of intersection A and B.

-10

-4

(0, 1)

B

A

-8

( 6, 9)

Put the compass at point A and draw an arc

2

2

y

4

below line . (Hint: Any compass setting

1

O

greater than

AB will work.)

2

x

P

-10

-4

(0, 1)

B

A

-8

( 6, 9)

Using the same compass setting, put the

y

3

3

4

compass at point B and draw an arc to

intersect the one drawn in step 2. Label

O

x

P

(0, 1)

the point of intersection Q.

-10

-4

B

Study Tip

Distance

Q

A

-8

Note that the distance

( 6, 9)

from a point to the x-axis

can be determined

by looking at the

Draw PQ. PQ

. Label point R at the intersection

y-coordinate and the

4

4

y

4

distance from a point

of PQ and .

Use the slopes of PQ and

to verify

to the y-axis can be

O

that the lines are perpendicular.

determined by looking

x

P

(0, 1)

-10

-4

B

The segment constructed from point P( 7,

2)

at the x-coordinate.

R

perpendicular to the line , appears to intersect

Q

line at R( 3,

5). Use the Distance Formula

A

-8

to find the distance between point P and line .

( 6, 9)

2

2

d

(x

x

)

(y

y

)

2

1

2

1

2

2

( 7

( 3))

( 2

( 5))

25 or 5

The distance between P and is 5 units.

DISTANCE BETWEEN PARALLEL LINES

Two lines in a plane are parallel if they are everywhere

A

B

C

D E

equidistant

. Equidistant means that the distance

between two lines measured along a perpendicular line

to the lines is always the same. The distance between

K

J

H

G

F

parallel lines is the length of the perpendicular segment

AK

BJ

CH

DG

EF

with endpoints that lie on each of the two lines.

160 Chapter 3 Parallel and Perpendicular Lines

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