Unit 13 – Perpendicular Lines
Day
Classwork
Day
Homework
Tuesday
4/18
Searching a Region in a Plane (d=rt)
1
HW 13.1
Wednesday
4/19
Parallel and Perpendicular Lines
2
HW 13.2
Thursday
4/20
Rotating 90° about a Point
3
HW 13.3
4
HW 13.4
5
HW 13.5
Equations of Medians and Perpendicular
Bisectors
Criterion for Perpendicularity
Friday
4/21
Standard Form of a Line
Normal Segments
Equations of Altitudes
Unit 13 Quiz 1
Monday
4/24
Applications of Parallel and Perpendicular
Lines
Distance from a point to a line
Distance between Parallel Lines
Tuesday
4/25
Dividing Segments Proportionally
6
HW 13.6
Wednesday
4/26
Review
7
Review Sheet
Study!
Unit 13 Quiz 2
Thursday
4/27
Review
8
Friday
4/28
UNIT 13 TEST
9
1
SEARCHING A REGION IN A PLANE [1]
Slope
Positive
Slope is used to describe the
measurement of steepness of
a straight line.
Slope is also described as
a _______________________________.
Negative
Undefined
Slope is a ratio that can be expressed as:
𝑚=
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
𝑜𝑟
Pythagorean Theorem
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
𝑜𝑟
𝑣𝑒𝑟𝑡 𝑐ℎ𝑎𝑛𝑔𝑒
ℎ𝑜𝑟𝑖𝑧 𝑐ℎ𝑎𝑛𝑔𝑒
𝑜𝑟
𝑦2− 𝑦1
𝑥2− 𝑥1
Distance Formula
Used to find the missing side in a right
triangle
Used to find the distance between 2 points
Distance = Rate x Time
Ex) If a car travels for 3 hours at 60 mph,
how far has the car traveled?
Time =
𝐃𝐢𝐬𝐭𝐚𝐧𝐜𝐞
Ex) How long will it take a car traveling at
60 mph to go 300 miles?
Rate =
𝐃𝐢𝐬𝐭𝐚𝐧𝐜𝐞
R
R
R
Zero
𝐑𝐚𝐭𝐞
𝐓𝐢𝐦𝐞
Ex) What is a car’s average rate of speed
if it has traveled 220 miles in 4 hours?
2
Consider the following:
Students in a robotics class must program a robot to move about an empty rectangular warehouse.
The program specifies location at a given time, 𝑡, seconds. The room is twice as long as it is wide.
Locations are represented as points in a coordinate plane with the southwest corner of the room
deemed the origin, (0,0), and the northeast corner deemed the point (2000 ft. , 1000 ft. ), as shown in
the diagram below.
(2000 ft. ,1000 ft. )
View of the Warehouse →
(0,0)
The first program written has the robot moving at a constant speed in a straight line. At time 𝑡 = 1
second, the robot is at position (30, 45), and at 𝑡 = 3 seconds, it is at position (50, 75). Answer the
questions below to program the robot’s motion.
1. Plot the given points on the smaller graph at the right.
2. Draw a line connecting the points.
3. How much did the 𝑥-coordinate change in 2 secs? In 1 sec?
4. How much did the 𝑦-coordinate change in 2 secs? In 1 sec?
5. At what time did the robot start its motion?
6. Where did the robot start its motion?
7. What is the ratio of change in 𝑦 to change in 𝑥? What does this rate of change represent?
8. What is the equation of the line of motion?
9. To the nearest foot, how far did the robot travel in 2 seconds?
3
10. What is the robot’s rate of speed?
(2000 ft. ,1000 ft. )
(0,0)
11. Which wall of the room will the robot hit?
12. Where is the location of impact (i.e. where will the robot hit the wall)?
13. What is the distance to the wall?
14. At what time will the robot hit the wall to the nearest tenth of a second?
4
PARALLEL AND PERPENDICULAR LINES [2]
Slopes of Perpendicular Lines
The lines 𝑙1 and 𝑙2 are perpendicular if and only
if their slopes are __________________________
________________________ of each other.
Ex: Find the slope of a line perpendicular to
y  2x  5
𝒍𝟏
(𝟏, 𝒎𝟏 )
(𝟏, 𝒎𝟐 )
𝒍𝟐
1) Write an equation of the line that passes through the point (7, -3) that intersects the line
𝟐𝒙 + 𝟓𝒚 = 𝟕 to form a right angle.
𝟑
2) Determine whether the lines given by the equations 𝟐𝒙 + 𝟑𝒚 = 𝟔 and 𝒚 = 𝟐 𝒙 + 𝟒 are
perpendicular. Support your answer.
3) Two lines having the same 𝒚-intercept are perpendicular. If the equation of one of these
𝟒
lines is 𝒚 = − 𝟓 𝒙 + 𝟔, what is the equation of the second line?
4) What is the relationship between two coplanar lines that are perpendicular to the same line?
5
5) Given two lines, 𝑙1 and 𝑙2, with equal slopes and a line 𝑘 that is perpendicular to one of these
two parallel lines, 𝑙1:
a) What is the relationship between line 𝑘 and the other line, 𝑙2 ?
b) What is the relationship between 𝑙1 and 𝑙2 ?
6) Given a point (−3, 6) and a line 𝑦 = 2𝑥 − 8:
a) What is the slope of the line?
b) What is the slope of any line parallel to the given line?
c) Write an equation of a line through the point and parallel to the line.
d) What is the slope of any line perpendicular to the given line? Explain.
1
7) Given the point (0, −7) and the line 𝑦 = 2 𝑥 + 5.
a) What is the slope of any line parallel to the given line? Explain your answer.
b) Write an equation of a line through the point and parallel to the line.
1
c) If a line is perpendicular to 𝑦 = 2 𝑥 + 5, will it be perpendicular to 𝑥 − 2𝑦 = 14? Explain.
6
1
8) Find an equation of a line through (√3, 2) and parallel to the line:
a) 𝑥 = −9
b) 𝑦 = −√7
c) What can you conclude about your answer in parts (a) and (b)?
9) Find an equation of a line through (−√2, 𝜋) parallel to the line 𝑥 − 7𝑦 = √5.
ROTATING 90° ABOUT A POINT & THE CRITERION FOR PERPENDICULARITY [3]
The image of a point rotated 90° (clockwise or counterclockwise) about any point, lies on the line
____________________________ to the original segment.
1) Plot the endpoints of segment AB: 𝐴(3, 7) to 𝐵(10, 1).
a) What is the slope of the original segment?
b) By rotating the segment 90°, what type of lines are
we creating?
c) What is the slope of the rotated segment?
d) Find the image of point B after a rotation of 90°
about point A.
e) Find the image of point B after a rotation of -90°
about point A.
7
2. Draw segment DE with endpoints D(1, 6) and E(-3, 5). Rotate the segment -90° about point E.
3. A triangle with endpoints 𝐴(2, 1), 𝐵(0, −4), and C(-5, 6) is rotated 90° about the point (0, −4). Find
the coordinates of the image.
4. Use the grid at the right showing points 𝑂(0,0), 𝑃(3, −1), and 𝑄(2,3) on the coordinate plane.
̅̅̅̅ and 𝑂𝑄
̅̅̅̅ are perpendicular using Pythagorean
Determine whether 𝑂𝑃
Theorem.
Support your findings using slope.
8
General Criterion for Perpendicularity
̅̅̅̅ that have a
Given two segments, ̅̅̅̅
𝑂𝐴 and 𝑂𝐵
A (x1, y1)
common endpoint at the origin 𝑂(0,0) and other
endpoints of 𝐴(𝑥1 , 𝑦1 ) and 𝐵(𝑥2 , 𝑦2 ).
̅̅̅̅
̅̅̅̅
𝑂𝐴 ⊥ 𝑂𝐵
iff
___________________________
B (x2, y2)
1) Plot the following points on a coordinate plane: 𝑂(0,0), 𝐴(6,4), and 𝐵(−2,3).
a) Draw ∆𝐴𝐵𝑂.
b) Is ∆𝐴𝐵𝑂 a right triangle? Justify your answer using slopes.
c) Justify your answer in part b using the general
criterion for perpendicularity.
Examples:
1) Carlos thinks that the segment having endpoint 𝐴(0, 0) and 𝐵 (6,0) is perpendicular to the
segment with endpoints 𝐴(0,0) and 𝐶(−2, 0). Do you agree? Why or why not?
̅̅̅̅ ⊥ 𝐴𝐵
̅̅̅̅ .
2) Given 𝐴(0,0) and 𝐵(3, −2), find possible coordinates of a point 𝐶 so that 𝐴𝐶
9
Writing the Equations of a Median and a Perpendicular Bisector
Recall: The median of a triangle is a segment drawn from a vertex to the ____________________ of
the opposite side.
A perpendicular bisector is a segment which is _______________________________ to a segment
at its _________________________.
y
1. Triangle PQR has coordinates A(2, -8), B(-4, 6) and C(8, 6).
a) Write an equation of the line containing the median
drawn from vertex A.
b) Write an equation of the line containing the median drawn from vertex B.
c) Write an equation of the line containing the median drawn from vertex C.
d) What are the coordinates of the centroid of the triangle?
10
y
2. Triangle PQR has coordinates D(3, 5), E(3, -1) and F(-5, -1).
a) Write an equation of the perpendicular bisector of DE .
b) Write an equation of the perpendicular bisector of EF .
c) Write an equation of the perpendicular bisector of DF .
d) What are the coordinates of the circumcenter of the triangle?
11
EQUATIONS OF LINES & NORMAL SEGMENTS [4]
Property of Normal Segments
A line segment with one endpoint on a line and perpendicular to
l
the line is called a normal segment to the line. A line containing
the normal segment is called the normal line.
Property of Perpendicular Lines
Given point 𝐴(𝑎, 𝑏) which lies on line 𝑙, point 𝐵(𝑐, 𝑑) not on line 𝑙,
with ̅̅̅̅
𝐴𝐵 ⊥ line 𝑙 , then any point 𝑃(𝑥, 𝑦) on line 𝑙 will satisfy the
̅̅̅̅ ⊥ 𝐴𝐵
̅̅̅̅ .
relationship 𝐴𝑃
y
1.) Given AB with coordinates 𝑨(𝟓, −𝟕) and 𝑩(𝟖, 𝟐).
a) Find an equation for the normal line through A.
b) Find an equation for the normal line through 𝐵.
2. Given the line 4 x  5 y  20 , find the equation of the normal segment passing through the origin.
3. Given the line 3x – 6y = 15, find the equation of the normal segment passing through the
point (8, -4).
12
Recall: An altitude of a triangle is a segment drawn from a vertex of the triangle
_____________________ to the line containing the opposite side. An altitude is a normal
segment.
y
4. Triangle PQR has coordinates P(-7, 8), Q(6, 3) and R(-2, -5).
a) Write an equation of the line containing the altitude
drawn from vertex P.
b) Write an equation of the line containing the altitude drawn from vertex Q.
c) Write an equation of the line containing the altitude drawn from vertex R.
13
y
5. Triangle PQR has coordinates A(4, 9), B(-8, 2) and C(7, 2).
a) Write an equation of the line containing the altitude
drawn from vertex A.
b) Write an equation of the line containing the altitude drawn from vertex B.
c) Write an equation of the line containing the altitude drawn from vertex C.
d) What are the coordinates of the orthocenter of the triangle?
14
APPLICATIONS OF PARALLEL AND PERPENDICULAR LINES [5]
1.) Quadrilateral LEAF has the coordinates L(2,5), E(7,1), A(2,-3), and F(-3,1). Prove quadrilateral
LEAF is a parallelogram.
2.) Is triangle RST, where R(1,5), S(5,1), and T(-1,-1), a right triangle? If so, which angle is the right
angle? Justify your answer.
3.) Consider the quadrilateral with vertices A(-2,-1), B(2,2), C(5,-2), and D(1,-5).
a) Show that the quadrilateral is a rectangle.
b) Is the quadrilateral a square? Explain.
15
Distance from a point to a line
If given a line and a point not on the line, then there exists exactly one line through the point that is
perpendicular to the given line.
Examples
1. Line l contains points at (-5, 3) and (4, -6). Find the distance between line l and point P(2, 4).
Distance from a Point to a Line
2.
Find the distance from x + y + 2 = 0 to (2, 4)
(Same question we just did without the formula.)
The distance from the point (x,y)
to the line Ax + By + C = 0
d=
3.
Ax + By + C
A2  B 2
Find the distance from (  1, 4) to 3x – 7y – 1 = 0
16
DISTANCE BETWEEN PARALLEL LINES
By definition, parallel lines do not intersect. An alternate definition states that two lines in a plane are
parallel if they are everywhere equidistant. Equidistant means that the distance between the two lines
measured along a perpendicular line to the lines is always the same.
In a plane, if two lines are each equidistant from a third line, then the two lines are parallel to each
other.
Example
a. Find the distance between the parallel lines a and b whose equations are 𝑥 + 3𝑦 = 6 and
𝑥 + 3𝑦 = −14, respectively.
17
Day 6—Dividing Segments Proportionally
a. Draw right triangle ABC.
b. What is the length of AC ?
c. What is the length of BC ?
d. Mark the halfway point on AC and
label it P. What are the coordinates of P?
e. Mark the halfway point on BC and
label it R. What are the coordinates of R?
f. Draw a segment from P to AB perpendicular to AC. Mark the intersection M. What are the
coordinates of M?
g. Draw a segment from R to AB perpendicular to BC. What do you notice?
h. Describe how you found point M.
Midpoint Formula
The midpoint formula can be used to find the
midpoint of a segment.
Finding the midpoint of a segment divides the
segment into equal parts. The segments have a
ratio of _____:_____
̅̅̅̅, for S(-6, 3) and T(1, 0).
1. Find the coordinates of M, the midpoint of 𝑆𝑇
̅ and L has coordinates (3, -5).
2. Find the coordinates of J if K(-1, 2) is the midpoint of 𝐽𝐿
18
3. a. Find the point, C, that partitions the segment, AB , with endpoints A(-3, 1) and B(5, 3), into a
ratio of 1:3.
b) What if we wanted to find the point that is 1 of the way along
4
segment
part a?
4.
AB,
closer to A than B? How is this the same/different as
Find the point on the directed segment from (-4,5) to (12,13) that divides it into a ratio of 1:7.
5a. Given points A(-4,5) and B(12,13), find the coordinates of the point C, that sits
segment
AB,
closer to A than it is to B.
b. Given points A(-4,5) and B(12,13), find the coordinates of the point D, that sits
segment
AB,
2
of the way along
5
2
of the way along
5
closer to B than it is to A.
19
6. Given points P(10,10) and Q(0,4), find point R on
PQ
such that
PR 7
 .
RQ 3
7. Given points A(3,-5) and B(19,-1), find the coordinates of point C that sits 3 of the way along
8
AB,
closer to A than to B.
20