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6. Contents Lesson 3-1Parallel Lines and Transversals Lesson 3-2Angles and Parallel Lines Lesson 3-3Slopes of Lines Lesson 3-4Equations of Lines Lesson 3-5Proving Lines Parallel Lesson 3-6Perpendiculars and Distance

7. Lesson 6 Contents Example 1Distance from a Point to a Line Example 2Construct a Perpendicular Segment Example 3Distance Between Lines OBJECTIVE: To find the distance between a point and a line and between parallel lines (2.9.8E) (M8.C.1)

8. Example 6-1a Draw the segment that represents the distance from Since 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, Answer:

9. Example 6-1b Answer: Draw the segment that represents the distance from

10. Example 6-2a Construct a line perpendicular to line s through V(1, 5) not on s. Then find the distance from V to s.

11. Example 6-2b Graph line s and point V. Place the compass point at point V. Make the setting wide enough so that when an arc is drawn, it intersects s in two places. Label these points of intersection A and B.

12. Example 6-2c Put the compass at point A and draw an arc below line s. (Hint: Any compass setting greater than will work.)

13. Example 6-2d Using the same compass setting, put the compass at point B and draw an arc to intersect the one drawn in step 2. Label the point of intersection Q.

14. Example 6-2e Draw. and s. Use the slopes of and s to verify that the lines are perpendicular.

15. Example 6-2f Answer: The distance between V and s is about 4.24 units. The segment constructed from point V(1, 5) perpendicular to the line s, appears to intersect line s at R(–2, 2). Use the Distance Formula to find the distance between point V and line s.

16. Example 6-2g Construct a line perpendicular to line m through Q(–4, –1) not on m. Then find the distance from Q to m. Answer:

17. Example 6-3a You will need to solve a system of equations to find the endpoints of a segment that is perpendicular to both a and b. The slope of lines a and b is 2. Find the distance between the parallel lines a and b whose equations are and respectively.

18. Example 6-3b Point-slope form Add 3 to each side. First, write an equation of a line p perpendicular to a and b. The slope of p is the opposite reciprocal of 2, Simplify. Use the y-intercept of line a, (0, 3), as one of the endpoints of the perpendicular segment.

19. Example 6-3c Next, use a system of equations to determine the point of intersection of line b and p. Substitute 2x–3 for y in the second equation.

20. Example 6-3d Substitute 2.4 for x in the equation for p. Simplify on each side. The point of intersection is (2.4, 1.8). Group like terms on each side.

21. Example 6-3e Distance Formula Then, use the Distance Formula to determine the distance between (0, 3) and (2.4, 1.8). Answer: The distance between the lines is or about 2.7 units.

22. Example 6-3f Answer: Find the distance between the parallel lines a and b whose equations are and respectively.

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