lesson 7: parallel, perpendicular lines and normal segments · lesson 7: parallel, perpendicular...
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Name _________________________ Period: _______ Date_____________
NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 7
GEOMETRY
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 point-
slope 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)
Name _________________________ Period: _______ Date_____________
NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 7
GEOMETRY
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 𝐴𝐵̅̅ ̅̅ is perpendicular to line 𝑙, and 𝐴𝑃̅̅ ̅̅ ⊥ 𝐴𝐵̅̅ ̅̅ ? Justify your answer. Plot the points on the coordinate grid.
We call segment 𝐴𝐵̅̅ ̅̅ 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 𝐴𝐵̅̅ ̅̅ . (normal line)
b. Find an equation for the line through 𝐵 and perpendicular to 𝐴𝐵̅̅ ̅̅ . (normal line)
Name _________________________ Period: _______ Date_____________
NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 7
GEOMETRY
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 2. Through (−5, −2), perpendicular to 𝑦 =5
2𝑥 + 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
Name _________________________ Period: _______ Date_____________
NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 7
GEOMETRY
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
3𝑥 − 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):
a. Write the equation of the normal segment to 𝑈𝑉̅̅ ̅̅ and goes through U
b. Write an equation for the line through 𝑉 and perpendicular to 𝑈𝑉̅̅ ̅̅ .
Name _________________________ Period: _______ Date_____________
NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 7
GEOMETRY
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?