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Tangent Line Construction with Compass and Straightedge

Learn how to construct tangent lines to circles using only a compass and straightedge. Discover key concepts and common errors in constructing tangent lines. Practice examples included.

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Tangent Line Construction with Compass and Straightedge

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  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. 3.3.1: Constructing Tangent Lines

  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. 3.3.1: Constructing Tangent Lines

  3. Key Concepts, continued 3.3.1: Constructing Tangent Lines

  4. Key Concepts, continued 3.3.1: Constructing Tangent Lines

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

  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. 3.3.1: Constructing Tangent Lines

  7. Key Concepts, continued 3.3.1: Constructing Tangent Lines

  8. Key Concepts, continued 3.3.1: Constructing Tangent Lines

  9. Key Concepts, continued 3.3.1: Constructing Tangent Lines

  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. 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. 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. 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 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. 3.3.1: Constructing Tangent Lines

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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