1. Mathematical Practices
Five-Minute Check (over Lesson 2–9)
Mathematical Practices
Then/Now
New Vocabulary
Key Concept: Distance Between a Point and a Line
Postulate 2.15: Perpendicular Postulate
Example 1: Real-World Example: Construct Distance From Point to a Line
Example 2: Distance from a Point to a Line on Coordinate Plane
Key Concept: Distance Between Parallel Lines
Theorem 2.25: Two Line Equidistant from a Third
Example 3: Distance Between Parallel Lines
Lesson Menu

2. Given 9  13, which segments are parallel?
A. AB || CD
B. FG || HI
C. CD || FG
D. none
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5-Minute Check 1

3. Given 2  5, which segments are parallel?
A. AB || CD
B. CD || FG
C. FG || HI
D. none
___ __
A. AB || CD
B. CD || FG
C. FG || HI
D. none
5-Minute Check 2

4. If m2 + m4 = 180, then AB || CD. What postulate supports this?
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A. If consecutive interior s are supplementary, lines are ||.
B. If alternate interior s are , lines are ||.
C. If corresponding s are , lines are ||.
D. If 2 lines cut by a transversal so that corresponding s are , then the lines are ||.
5-Minute Check 3

5. If 5  14, then CD || HI. What postulate supports this?__
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A. If corresponding s are , lines are ||.
B. If 2 lines are to the same line, they are ||.
C. If alternate interior s are , lines are ||.
D. If consecutive interior s are supplementary, lines are ||. ┴
5-Minute Check 4

6. Find x so that AB || HI if m1 = 4x + 6 and m14 = 7x – 27. .__
___ .
A. 6.27
B. 11 C
D. 18
5-Minute Check 5

7. Two lines in the same plane do not intersect
Two lines in the same plane do not intersect. Which term best describes the relationship between the lines?
A. parallel
B. perpendicular
C. skew
D. transversal
5-Minute Check 6

8. Mathematical Practices 2 Reason abstractly and quantitatively.
4 Model with mathematics.
Content Standards
G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). MP

9. Find the distance between a point and a line.
You proved that two lines are parallel using angle relationships.
Find the distance between a point and a line.
Find the distance between parallel lines.
Then/Now

10. equidistant
Vocabulary

11. Concept

12. Concept

13. Construct Distance from a Point to a Line
CONSTRUCTION A certain roof truss is designed so that the center post extends from the peak of the roof (point A) to the main beam. Construct and name the segment whose length represents the shortest length of wood that will be needed to connect the peak of the roof to the main beam.
The distance from a line to a point not on the line is the length of the segment perpendicular to the line from the point. Locate points R and S on the main beam equidistant from point A.
Example 1

14. Construct Distance From Point to a Line
Locate a second point not on the beam equidistant from R and S. Construct AB so that AB is perpendicular to the beam.
Answer: The measure of AB represents the shortest length of wood needed to connect the peak of the roof to the main beam.
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Example 1

15. KITES Which segment represents the shortest distance from point A to DB?
A. AD
B. AB
C. CX
D. AX
Example 1

16. Step 1 Find the slope of line s.
Distance from a Point to a Line on a Coordinate Plane
COORDINATE GEOMETRY Line s contains points at (0, 0) and (–5, 5). Find the distance between line s and point V(1, 5).
(–5, 5)
(0, 0)
V(1, 5)
Step 1 Find the slope of line s.
Begin by finding the slope of the line through points (0, 0) and (–5, 5).
Example 2

17. The equation of line s is y = –x.
Distance from a Point to a Line on a Coordinate Plane
Then write the equation of this line by using the point (0, 0) on the line.
Slope-intercept form
m = –1, (x1, y1) = (0, 0)
Simplify.
The equation of line s is y = –x.
Example 2

18. Subtract 1 from each side.
Distance from a Point to a Line on a Coordinate Plane
Step 2 Write an equation of the line t perpendicular to line s through V(1, 5).
Since the slope of line s is –1, the slope of line t is 1. Write the equation for line t through V(1, 5) with a slope of 1.
Slope-intercept form
m = 1, (x1, y1) = (1, 5)
Simplify.
Subtract 1 from each side.
The equation of line t is y = x + 4.
Example 2

19. 2y = 4 Add the two equations. y = 2 Divide each side by 2.
Distance from a Point to a Line on a Coordinate Plane
Step 3 Solve the system of equations to determine the point of intersection.
line s: y = –x
line t: (+) y = x + 4
2y = 4 Add the two equations.
y = 2 Divide each side by 2.
Solve for x.
2 = –x Substitute 2 for y in the first equation.
–2 = x Divide each side by –1.
The point of intersection is (–2, 2). Let this point be Z.
Example 2

20. Distance from a Point to a Line on a Coordinate Plane
Step 4 Use the Distance Formula to determine the distance between Z(–2, 2) and V(1, 5).
Distance formula
Substitution
Simplify.
Answer: The distance between the point and the line is or about 4.24 units.
Example 2

21. COORDINATE GEOMETRY Line n contains points (2, 4) and (–4, –2)
COORDINATE GEOMETRY Line n contains points (2, 4) and (–4, –2). Find the distance between line n and point B(3, 1).
B(3, 1)
(2, 4)
(–4, 2) A. B. C. D.
Example 2

22. Concept

23. Concept

24. Distance Between Parallel Lines
Find the distance between the parallel lines a and b with equations y = 2x + 3 and y = 2x − 1, respectively.
You will need to solve a system of equations to find the endpoints of a segment that is perpendicular to both a and b. From their equations, we know that the slope of line a and line b is 2.
Sketch line p through the y-intercept of line b, (0, –1), perpendicular to lines a and b. a b p
Example 3

25. Subtract 1 from each side.
Distance Between Parallel Lines
Step 1
Write an equation for line p. The slope of p is the opposite reciprocal of
Use the y-intercept of line b, (0, –1), as one of the endpoints of the perpendicular segment.
Point-slope form
Simplify.
Subtract 1 from each side.
Example 3

26. Substitute 2x + 3 for y in the second equation.
Distance Between Parallel Lines
Step 2
Use a system of equations to determine the point of intersection of the lines a and p.
Substitute 2x + 3 for y in the second equation.
Group like terms on each side.
Example 3

27. Substitute for x in the equation for p.
Distance Between Parallel Lines
Simplify on each side.
Multiply each side by .
Substitute for x in the equation for p.
Example 3

28. The point of intersection is or (–1.6, –0.2).
Distance Between Parallel Lines
Simplify.
The point of intersection is or (–1.6, –0.2).
Example 3

29. Answer: The distance between the lines is about 1.79 units.
Distance Between Parallel Lines
Step 3
Use the Distance Formula to determine the distance between (0, –1) and (–1.6, –0.2).
Distance Formula
x2 = –1.6, x1 = 0, y2 = –0.2, y1 = –1
Answer: The distance between the lines is about 1.79 units.
Example 3

30. Find the distance between the parallel lines a and b whose equations are and , respectively.
A units
B units
C units
D. 3 units
Example 3