M2 GEOMETRY PACKET 6 OF UNIT 2 – SECTION 3-6 DISTANCES TO LINES
ASSIGNMENTS FOR PACKET 6 OF UNIT 2
DUE
NUMBER
ASSIGNMENT
2N
p. 5-6 in this packet
2O
p. 10-11 in this packet
1
TOPICS
3-6:
Find the distance between two
parallel lines.
3-6:
Find the distance from a point to a
line.
M2 GEOMETRY PACKET 6 OF UNIT 2 – SECTION 3-6 DISTANCES TO LINES
Distance Between Parallel Lines
The distance between two parallel lines is the length of any perpendicular segment with an endpoint on each
line.
Read the example below:
and m with the equations
Find the distance between parallel lines
y  2 x  1 and y  2 x  4 .
Step 1: Find the y-intercept of line ℓ.
Line ℓ has y-intercept 1.
Step 2: Draw line p so that it is perpendicular to ℓ and m and has the same yintercept as line ℓ.
Step 3: Write an equation for line p.
1
Line p has slope
and y-intercept 1.
2
1
x  1.
An equation of p is y 
2
Step 4: Find the point at which p intersects m.
To find the intersection, solve a system of equations.
Line m : y  2 x  4
Line p : y 
1
x 1
2
1
x 1
2
Substitute:
2x  4 
Multiply by 2 to clear out fractions:
4x  8   x  2
5x  10
Add x:
x2
Divide by 2:
Substitute 2 for x to find the y-coordinate:
The intersection of p and m is  2, 0  .
y  2x  4  y  2  2  4  0
.
Step 5: Use the Distance Formula to find the length of the segment from  0,1 to  2, 0  .
d

 x2  x1    y2  y1 
2
 2  0    0  1
2
2
2
 5
The distance between ℓ and m is
2
5 units.
M2 GEOMETRY PACKET 6 OF UNIT 2 – SECTION 3-6 DISTANCES TO LINES
Follow the steps to find the distance between each pair of parallel lines.
1.
: y  x  3 , m : y  x 1
a. Graph both lines.
b. Find the y-intercept of line
. Call this point A.
c. Draw line p so that it is perpendicular to
and has the same y-intercept as .
and m
d. Write an equation for line p.
e. Write the equations of m and p below, and use substitution to find their intersection point B.
m:
Intersection point B: (
p:
,
)
f. Use the Distance Formula to find the length of AB . Leave your answer in simplest radical form. This is
the distance between the parallel lines.
Distance =
3
M2 GEOMETRY PACKET 6 OF UNIT 2 – SECTION 3-6 DISTANCES TO LINES
Follow the steps to find the distance between each pair of parallel lines.
2.
: y  2 x , m : y  2 x  5
a. Graph both lines.
b. Find the y-intercept of line
. Call this point A.
c. Draw line p so that it is perpendicular to
and has the same y-intercept as .
and m
d. Write an equation for line p.
e. Write the equations of m and p below, and use substitution to find their intersection point B.
m:
Intersection point B: (
p:
,
)
f. Use the Distance Formula to find the length of AB . Leave your answer in simplest radical form. This is
the distance between the parallel lines.
Distance =
4
M2 GEOMETRY PACKET 6 OF UNIT 2 – SECTION 3-6 DISTANCES TO LINES
ASSIGNMENT 2N – DISTANCE BETWEEN TWO PARALLEL LINES
Follow the steps to find the distance between each pair of parallel lines.
1.
: y  3x , m : y  3x  10
a. Graph both lines.
b. Find the y-intercept of line
. Call this point A.
c. Draw line p so that it is perpendicular to
and has the same y-intercept as .
and m
d. Write an equation for line p.
e. Write the equations of m and p below, and use substitution to find their intersection point B.
m:
Intersection point B: (
p:
,
)
f. Use the Distance Formula to find the length of AB . Leave your answer in simplest radical form. This is
the distance between the parallel lines.
Distance =
5
M2 GEOMETRY PACKET 6 OF UNIT 2 – SECTION 3-6 DISTANCES TO LINES
ASSIGNMENT 2N (CONTINUED)
Follow the steps to find the distance between each pair of parallel lines.
2.
: y  2 x  5 , m : y  2 x  5
a. Graph both lines.
b. Find the y-intercept of line
. Call this point A.
c. Draw line p so that it is perpendicular to
and has the same y-intercept as .
and m
d. Write an equation for line p.
e. Write the equations of m and p below, and use substitution to find their intersection point B.
m:
Intersection point B: (
p:
,
)
f. Use the Distance Formula to find the length of AB . Leave your answer in simplest radical form. This is
the distance between the parallel lines.
Distance =
6
M2 GEOMETRY PACKET 6 OF UNIT 2 – SECTION 3-6 DISTANCES TO LINES
DISTANCE FROM A POINT TO A LINE
How do we calculate the distance from a point to a line?
Check off steps as you complete them:
______ Open Geogebra, and select Geometry.
______ Draw AB and point C, not on the line.
______ Change the line tool to Segment, and draw a segment
from C to AB . This should create a new point D on AB .
______ Measure the length of CD using the Distance or Length command. (See above.)
______ Click on the 3 horizontal lines in the upper right corner.
Click Options, Rounding, and select 4 Decimal Places.
______ Drag D until the length of CD is as small as you can
make it.
______ Select Angle (see picture above right), and click on
points B, D, and C in that order to measure BDC .
What do you notice about the angle between CD and AB ?
Complete this statement:
Distance From a Point to a Line When a point is not on a line, the distance from the point to
the line is the length of the segment that contains the point and is ________________________
to the line.
7
M2 GEOMETRY PACKET 6 OF UNIT 2 – SECTION 3-6 DISTANCES TO LINES
Follow the steps to find the distance from point P to line .
1. Line
contains points  0, 2  and  6,6  . Point P has coordinates  1,5 .
a. Graph line
and point P.
b. Find the slope of line
for line .
, and write an equation
c. Find the slope of a line perpendicular to
.
d. Graph the line perpendicular to and passing
through point P. Call this line k, and write its
equation below.
Equation of line k:
e. Use substitution to find the intersection point Q of lines
Intersection point Q: (
,
and k.
)
f. Use the distance formula to find the distance between P and Q. Write your answer in simplest radical
form. This is the distance between line and point P.
Distance =
8
M2 GEOMETRY PACKET 6 OF UNIT 2 – SECTION 3-6 DISTANCES TO LINES
Follow the steps to find the distance from point P to line .
2. Line
contains points  2, 4  and  5,1 . Point P has coordinates 1,1 .
a. Graph line
and point P.
b. Find the slope of line
for line .
, and write an equation
c. Find the slope of a line perpendicular to
.
d. Graph the line perpendicular to and passing
through point P. Call this line k, and write its
equation below.
Equation of line k:
e. Use substitution to find the intersection point Q of lines
Intersection point Q: (
,
and k.
)
f. Use the distance formula to find the distance between P and Q. Write your answer in simplest radical
form. This is the distance between line and point P.
Distance =
9
M2 GEOMETRY PACKET 6 OF UNIT 2 – SECTION 3-6 DISTANCES TO LINES
ASSIGNMENT 2O – DISTANCE FROM A POINT TO A LINE
Follow the steps to find the distance from point P to line .
1. Line
contains points  4, 2  and  2,0  . Point P has coordinates  3,7  .
a. Graph line
and point P.
b. Find the slope of line
for line .
, and write an equation
c. Find the slope of a line perpendicular to
.
d. Graph the line perpendicular to and passing
through point P. Call this line k, and write its
equation below.
Equation of line k:
e. Use substitution to find the intersection point Q of lines
Intersection point Q: (
,
and k.
)
f. Use the distance formula to find the distance between P and Q. Write your answer in simplest radical
form. This is the distance between line and point P.
Distance =
10
M2 GEOMETRY PACKET 6 OF UNIT 2 – SECTION 3-6 DISTANCES TO LINES
ASSIGNMENT 2O (CONTINUED)
Follow the steps to find the distance from point P
to line .
2. Line
contains points  7,8 and  0,5 . Point
P has coordinates  5,32  .
a. Sketch a graph of line
and point P.
b. Find the slope of line
equation for line .
, and write an
c. Find the slope of a line perpendicular to
d. Graph the line perpendicular to
below.
.
and passing through point P. Call this line k, and write its equation
Equation of line k:
e. Use substitution to find the intersection point Q of lines
Intersection point Q: (
,
and k.
)
f. Use the distance formula to find the distance between P and Q. Write your answer in simplest radical
form. This is the distance between line and point P.
Distance =
11
M2 GEOMETRY PACKET 6 OF UNIT 2 – SECTION 3-6 DISTANCES TO LINES
REVIEW OF SECTION 3-6 DISTANCES TO LINES
1.
: y  2x  7 , m : y  2x  3
a. Graph both lines.
b. Find the y-intercept of line
. Call this point A.
c. Draw line p so that it is perpendicular to
and has the same y-intercept as .
and m
d. Write an equation for line p.
e. Use substitution to find the intersection B of lines m and p.
m:
Intersection point B: (
p:
,
)
f. Find the distance between the parallel lines. Leave your answer in simplest radical form.
Distance =
12
M2 GEOMETRY PACKET 6 OF UNIT 2 – SECTION 3-6 DISTANCES TO LINES
2.
: y  3x  12 , m : y  3x 18
a. Sketch the graph of both lines.
b. Find the y-intercept of line
. Call this point A.
c. Draw line p so that it is perpendicular to
m and has the same y-intercept as .
and
d. Write an equation for line p.
e. Use substitution to find the intersection B of
lines m and p.
m:
Intersection point B: (
p:
,
)
f. Find the distance between the parallel lines. Leave your answer in simplest radical form.
Distance =
13
M2 GEOMETRY PACKET 6 OF UNIT 2 – SECTION 3-6 DISTANCES TO LINES
3. Line
contains points  2, 4  and 1, 9  . Point P has coordinates 14, 6  .
a. Sketch a graph of line
and point P.
b. Find the slope of line
equation for line .
, and write an
c. Find the slope of a line perpendicular to
.
d. Graph the line perpendicular to and
passing through point P. Call this line k, and
write its equation below.
Equation of line k:
e. Use substitution to find the intersection Q of lines
Intersection point Q: (
,
f. Find the distance between line
and k.
)
and point P. Write your answer in simplest radical form.
Distance =
14
M2 GEOMETRY PACKET 6 OF UNIT 2 – SECTION 3-6 DISTANCES TO LINES
4. Line
contains points  2,0  and  4,8 . Point P has coordinates  5,1 .
a. Graph line
and point P.
b. Find the slope of line
line .
, and write an equation for
c. Find the slope of a line perpendicular to
.
d. Graph the line perpendicular to and passing
through point P. Call this line k, and write its
equation below.
Equation of line k:
e. Use substitution to find the intersection Q of lines
Intersection point Q: (
,
f. Find the distance between line
and k.
)
and point P. Write your answer in simplest radical form.
Distance =
15