1. Introduction Tangent lines are useful in calculating distances as well as diagramming in the professions of construction, architecture, and landscaping. Geometry construction tools can be used to create lines tangent to a circle. As with other constructions, the only tools you are allowed to use are a compass and a straightedge, a reflective device and a straightedge, or patty paper and a straightedge. You may be tempted to measure angles or lengths, but remember, this is not allowed with constructions. 1 3.3.1: Constructing Tangent Lines

2. 2 Key Concepts If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency, the only point at which a line and a circle intersect. Exactly one tangent line can be constructed by using construction tools to create a line perpendicular to the radius at a point on the circle.

3. Key Concepts, continued 3 3.3.1: Constructing Tangent Lines Constructing a Tangent at a Point on a Circle Using a Compass 1.Use a straightedge to draw a ray from center O through the given point P. Be sure the ray extends past point P. 2.Construct the line perpendicular to at point P. This is the same procedure as constructing a perpendicular line to a point on a line. a.Put the sharp point of the compass on P and open the compass less wide than the distance of. b.Draw an arc on both sides of P on. Label the points of intersection A and B. (continued)

4. Key Concepts, continued 4 3.3.1: Constructing Tangent Lines c.Set the sharp point of the compass on A. Open the compass wider than the distance of and make a large arc. d.Without changing your compass setting, put the sharp point of the compass on B. Make a second large arc. It is important that the arcs intersect each other. 3.Use your straightedge to connect the points of intersection of the arcs. 4.Label the new line m. Do not erase any of your markings. Line m is tangent to circle O at point P.

5. Key Concepts, continued It is also possible to construct a tangent line from an exterior point not on a circle. 5 3.3.1: Constructing Tangent Lines

6. 6 Key Concepts, continued If two segments are tangent to the same circle, and originate from the same exterior point, then the segments are congruent.

7. Key Concepts, continued 7 3.3.1: Constructing Tangent Lines Constructing a Tangent from an Exterior Point Not on a Circle Using a Compass 1.To construct a line tangent to circle O from an exterior point not on the circle, first use a straightedge to draw a ray connecting center O and the given point R. 2.Find the midpoint of by constructing the perpendicular bisector. a.Put the sharp point of your compass on point O. Open the compass wider than half the distance of b.Make a large arc intersecting. (continued)

8. Key Concepts, continued 8 3.3.1: Constructing Tangent Lines c.Without changing your compass setting, put the sharp point of the compass on point R. Make a second large arc. It is important that the arcs intersect each other. Label the points of intersection of the arcs as C and D. d.Use your straightedge to connect points C and D. e.The point where intersects is the midpoint of. Label this point F. (continued)

9. Key Concepts, continued 9 3.3.1: Constructing Tangent Lines 3.Put the sharp point of the compass on midpoint F and open the compass to point O. 4.Without changing the compass setting, draw an arc across the circle so it intersects the circle in two places. Label the points of intersection as G and H. 5.Use a straightedge to draw a line from point R to point G and a second line from point R to point H. Do not erase any of your markings. and are tangent to circle O.

10. Key Concepts, continued If two circles do not intersect, they can share a tangent line, called a common tangent. Two circles that do not intersect have four common tangents. Common tangents can be either internal or external. 10 3.3.1: Constructing Tangent Lines

11. Key Concepts, continued A common internal tangent is a tangent that is common to two circles and intersects the segment joining the radii of the circles. 11 3.3.1: Constructing Tangent Lines

12. Key Concepts, continued A common external tangent is a tangent that is common to two circles and does not intersect the segment joining the radii of the circles. 12 3.3.1: Constructing Tangent Lines

13. Common Errors/Misconceptions assuming that a radius and a line are perpendicular at the possible point of intersection simply by observation assuming two tangent lines are congruent by observation incorrectly changing the compass settings not making large enough arcs to find the points of intersection 13 3.3.1: Constructing Tangent Lines

14. Guided Practice Example 1 Use a compass and a straightedge to construct tangent to circle A at point B. 14 3.3.1: Constructing Tangent Lines

15. Guided Practice: Example 1, continued 1.Draw a ray from center A through point B and extending beyond point B. 15 3.3.1: Constructing Tangent Lines

16. Guided Practice: Example 1, continued 2.Put the sharp point of the compass on point B. Set it to any setting less than the length of, and then draw an arc on either side of B, creating points D and E. 16 3.3.1: Constructing Tangent Lines

17. Guided Practice: Example 1, continued 17 3.3.1: Constructing Tangent Lines

18. Guided Practice: Example 1, continued 3.Put the sharp point of the compass on point D and set it to a width greater than the distance of. Make a large arc intersecting. 18 3.3.1: Constructing Tangent Lines

19. Guided Practice: Example 1, continued 19 3.3.1: Constructing Tangent Lines

20. Guided Practice: Example 1, continued 4.Without changing the compass setting, put the sharp point of the compass on point E and draw a second arc that intersects the first. Label the point of intersection with the arc drawn in step 3 as point C. 20 3.3.1: Constructing Tangent Lines

21. Guided Practice: Example 1, continued 21 3.3.1: Constructing Tangent Lines

22. Guided Practice: Example 1, continued 5.Draw a line connecting points C and B, creating tangent. Do not erase any of your markings. is tangent to circle A at point B. 22 3.3.1: Constructing Tangent Lines

23. Guided Practice: Example 1, continued 23 3.3.1: Constructing Tangent Lines ✔

24. Guided Practice: Example 1, continued 24 3.3.1: Constructing Tangent Lines

25. Guided Practice Example 3 Use a compass and a straightedge to construct the lines tangent to circle C at point D. 25 3.3.1: Constructing Tangent Lines

26. Guided Practice: Example 3, continued 1.Draw a ray connecting center C and the given point D. 26 3.3.1: Constructing Tangent Lines

27. Guided Practice: Example 3, continued 2.Find the midpoint of by constructing the perpendicular bisector. Put the sharp point of your compass on point C. Open the compass wider than half the distance of. Make a large arc intersecting. 27 3.3.1: Constructing Tangent Lines

28. Guided Practice: Example 3, continued 28 3.3.1: Constructing Tangent Lines

29. Guided Practice: Example 3, continued Without changing your compass setting, put the sharp point of the compass on point D. Make a second large arc. It is important that the arcs intersect each other. Label the points of intersection of the arcs as E and F. 29 3.3.1: Constructing Tangent Lines

30. Guided Practice: Example 3, continued 30 3.3.1: Constructing Tangent Lines

31. Guided Practice: Example 3, continued Use your straightedge to connect points E and F. The point where intersects is the midpoint of. Label this point G. 31 3.3.1: Constructing Tangent Lines

32. Guided Practice: Example 3, continued 32 3.3.1: Constructing Tangent Lines

33. Guided Practice: Example 3, continued 3.Put the sharp point of the compass on midpoint G and open the compass to point C. Without changing the compass setting, draw an arc across the circle so it intersects the circle in two places. Label the points of intersection as H and J. 33 3.3.1: Constructing Tangent Lines

34. Guided Practice: Example 3, continued 34 3.3.1: Constructing Tangent Lines

35. Guided Practice: Example 3, continued 4.Use a straightedge to draw a line from point D to point H and a second line from point D to point J. Do not erase any of your markings. and are both tangent to circle C. 35 3.3.1: Constructing Tangent Lines

36. Guided Practice: Example 3, continued 36 3.3.1: Constructing Tangent Lines ✔

37. Guided Practice: Example 3, continued 37 3.3.1: Constructing Tangent Lines