GEOMETRY
NYS COMMON CORE MATHEMATICS CURRICULUM
Name _________________________
Lesson 7
M4
Period: _______ Date_____________
Lesson 7: Parallel, Perpendicular Lines and Normal Segments
Warm up
Write the equations of lines with the following characteristics in either slope-intercept 𝑦 = 𝑚𝑥 + 𝑏 or pointslope 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) form.
1. Slope = −2, 𝑦-intercept (0, −4)
2. Passing through points (−1, −5) and (3, 3)
3. A vertical line that passes through (−4, 5)
4. 𝑦-intercept (0, −4) and passes through
(3, −6)
5. Passing through (1, −6) and (0, 3)
6. A vertical line that passes through (−4, 5)
GEOMETRY
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
Name _________________________
M4
Period: _______ Date_____________
Lesson 7: Parallel, Perpendicular Lines and Normal Segments
Learning Target: I can write the equations of lines parallel and perpendicular to a given line
Opening Activity:
Given points 𝐴(3,4) and 𝑃(5,10) which lie on line 𝑙, and
point 𝐵(6, 3) not on line 𝑙, can we say that ̅̅̅̅
𝐴𝐵 is
̅̅̅̅
̅̅̅̅
perpendicular to line 𝑙, and 𝐴𝑃 ⊥ 𝐴𝐵 ? Justify your answer.
Plot the points on the coordinate grid.
We call segment ̅̅̅̅
𝐴𝐵 a normal segment to line 𝑙 because it has
one endpoint on the line and is perpendicular to the line.
Definition: A line segment with one endpoint on a line and
perpendicular to the line is called a ____________________
_______________ to the line.
Example 1. Given 𝐴(5, −7) and 𝐵(8, 2):
a. Find an equation for the line through 𝐴 and perpendicular to ̅̅̅̅
𝐴𝐵 . (normal line)
b. Find an equation for the line through 𝐵 and perpendicular to ̅̅̅̅
𝐴𝐵 . (normal line)
GEOMETRY
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
Name _________________________
M4
Period: _______ Date_____________
Example 2. Write the equation of the line described in either slope-intercept 𝑦 = 𝑚𝑥 + 𝑏 or point-slope 𝑦 −
𝑦1 = 𝑚(𝑥 − 𝑥1 ) form.
1. Through (−3, −5), parallel to 𝑦 = −4𝑥 + 3
5
2. Through (−5, −2), perpendicular to 𝑦 = 2 𝑥 + 2
Example 3. Write the equation of a line perpendicular to 𝑦 = 𝑥 − 2 and passes through (2, −1)
Example 4. Write the equation of a line that is parallel to the line whose equation is to 2𝑦 − 10 = 𝑥
Example 5. Are the lines to 3𝑦 − 𝑥 = 3 and
𝑦 + 3 = 3(𝑥 − 1) perpendicular, parallel or neither?
Example 6. Find the equation of a line that is perpendicular to the given line and has the same y-intercept
GEOMETRY
NYS COMMON CORE MATHEMATICS CURRICULUM
Name _________________________
M4
Lesson 7
Period: _______ Date_____________
Lesson 7: Parallel, Perpendicular Lines and Normal Segments
Classwork
Write the equation of the line described in either form:
slope-intercept 𝑦 = 𝑚𝑥 + 𝑏 or point-slope 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
1. Find the equation of a line that passes through (4, −2) and is perpendicular to 𝑦 =
1
3
𝑥−3
2. Find the equation of a line that passes through (5, −4) and parallel to 5𝑦 = 3𝑥 − 4
3. Find the equation of a line that passes through (−3, 3) and perpendicular to 5𝑦 = 3𝑥 − 4
4. Find the equation of a line that passes through (−1, −2) and is parallel to 𝑦 − 4𝑥 = −1
5. Given 𝑈(−4, −1) and 𝑉(7, 1):
̅̅̅̅ and goes through U
a. Write the equation of the normal segment to 𝑈𝑉
̅̅̅̅.
b. Write an equation for the line through 𝑉 and perpendicular to 𝑈𝑉
GEOMETRY
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
Name _________________________
M4
Period: _______ Date_____________
6. Write the equation of a line that is perpendicul to the given line has the
same y-intercept.
7. Determine whether the two lines represented by the equations 𝑦 = 2𝑥 + 3 and 2𝑦 + 𝑥 = 6 are
parallel, perpendicular, or neither. Justify your response.
8. Two lines are represented by the equations 𝑥 + 2𝑦 = 4 and 4𝑦 − 2𝑥 = 12 . Determine whether
these lines are parallel, perpendicular, or neither. Justify your answer.
9. Two parallel roads run through a town. When the roads are graphed on the coordinate plane, one of the roads
can be represented by the equation 2𝑥 + 3𝑦 = 6. If the other road passes through the point (6,7), what is
the equation of the second road?
10. Find the equation of a line perpendicular to 4𝑥 − 𝑦 = −4 and shares the same y-intercept
11. What is the distance from point (3, −1) to (5, 4) on the coordinate plane?