1. Glencoe Geometry Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Developed by FSCreations, Inc., Cincinnati, Ohio 45202 Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240 3.6 Perpendiculars and Distance

2. Distance from a Point to a Line The distance from a line to a point not on the line is the length of the segment ┴ to the line from the point. l A

3. 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: Example 1:

4. Turn to Pages 162 – 163 in your textbook and complete #11 – 16. Your Turn:

5. Constructing a ┴ Segment How do we construct a ┴ segment accurately? How do we use a compass? By using a compass. The next example will show us.

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

7. 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. Example 2:

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

9. 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. Example 2:

10. Draw. and s. Use the slopes of and s to verify that the lines are perpendicular. Example 2:

11. 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. Example 2:

12. Turn to Page 163 in your textbook and complete #17 – 18. Your Turn:

14. Distance Between Parallel Lines Two lines in a plane are || if they are equidistant everywhere. To verify if two lines are equidistant find the distance between the two || lines by calculating the distance between one of the lines and any point on the other line.

15. Theorem 3.9 In a plane, if two lines are equidistant from a third line, then the two lines are || to each other.

16. 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. Example 3:

17. 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. Example 3:

18. 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. Example 3:

19. 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. Example 3:

20. 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. Example 3:

21. Turn to Page 163 in your textbook and complete #19 – 22. Pre-AP: Add #24 Your Turn: