2.8 Distance from a Point to a Line

Distance from a point to a nonvertical line is

Ex 1) Find distance between line & point (nearest hundredth) a) y = 3x + 2 P(3, 4)

point (x1, y1)

(notice all on one side!)

3x – y + 2 = 0

line equation: Ax + By + C = 01 1

2 2

Ax By Cd

A B

2 2

3(3) ( 1)(4) 2 9 4 2

9 1(3) ( 1)d

7 72.21

10 10

Ex 1) Find distance between line & point (nearest hundredth) b) Line through (4, 2) and (–4, 5) and the point (4, 0) *have to find equation of line first!

2 2

(3)(4) (8)(0) 28 12 0 28 16 161.87

9 64 73 73(3) (8)d

2 52 ( 4)

4 4y x

32 ( 4)

88 16 3 12

3 8 28 0

y x

y x

x y

To find the distance between 2 parallel lines, simply find a point on one line first and then use the second line for equation

Ex 2) Find the distance between parallel?

1 12 24 and 3y x y x

a point?

2 2

(1)(0) ( 2)( 3) 8 0 6 8 146.26

1 4 5(1) ( 2)d

122( 4)

2 8

2 8 0

y x

y x

x y

yep!

How about (0, –3)

12 3y x

2 2

(.75)(0) (3)(7.4) 27 4.81.6 ft

9.5625(.75) (3)d

These questions have lots of real world applicatons.Advice Draw a picture!

Ex 3) An aluminum awning has a wrought iron support bar attached to the building & the awning so that it is perpendicular to the awning. From the picture, determine the length of the support bar (to nearest tenth)

9 8.259 ( 0)

0 3.75

93

3 27 .75

.75 3 27 0

y x

y x

y x

x y

equation:(3, 8.25)(0, 9)

(0, 7.4)

(wall)

point: (0, 7.4)

We can indirectly measure lengths of things like wells by using gravity constant & speed of sound

Because of gravity: d = 16t12 (t1 = time down)

Because of sound: d = 1100t2 (t2 = time of sound back up)

so 16t12 = 1100t2

Total time: T = t1 + t2 or t2 = T – t1

substitute…. 16t12 = 1100(T – t1)

16t12 + 1100t1 – 1100T = 0

Ex 4) A rock was dropped in a well. The total time T for the rock to reach the bottom of the well and the sound to travel back was 4.5s. Find the depth of the well to the nearest foot.

16t12 + 1100t1 – 1100(4.5) = 0

16t12 + 1100t1 – 4950 = 0

use quad. form. 2

1

1100 (1100) 4(16)( 4950)

2(16)t

t1 = 4.24 or –72.99

so, d = 16t12 = 16(4.24)2 = 288 ft.

Ex 5) Find the equations of all lines parallel to 3x – 5y = 0 located 2 units from the line.

2 2

3 52

(3) ( 5)

x y

1 1

2 2

Ax By Cd

A B

d = ± 2(why ±? Abs value means it can be +2 or –2)

3 52

34

x y

2 34 3 5

3 5 2 34 0

x y

x y

so

Homework

#209 Pg 111 #1–9 all 12, 14, 16, 18, 20, 21, 24, 25, 28