1 / 36

Tangents

Tangents. Dr. Mohamed BEN ALI. Objectives. By the end of this lecture, students will be able to: Understand the types of Tangents. Construct tangents. Construct incircle . Construct circumcircle. Some definitions.

marged
Download Presentation

Tangents

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Tangents Dr. Mohamed BEN ALI

  2. Objectives • By the end of this lecture, students will be able to: • Understand the types of Tangents. • Construct tangents. • Construct incircle. • Construct circumcircle

  3. Some definitions • Centre - the point within the circle where the distance to points on the circumference is the same. • radius - the distance from the centre to any point on the circle. The diameter is twice the radius. • circumference(perimeter) - the distance around a circle. • diameter- a chord(of max. length) passing through the centre

  4. Some definitions • chord is a straight line joining two points on the circumference. • If line intersect the circle at two point that is called secant • tangent - a straight line making contact at one point on the circumference, such that the radius from the centre is at right angles to the line.

  5. Drawing a Tangent-line to a Circle • Given a circle, and any point on it, we can draw a straight line through the point that will be tangent to this circle

  6. How do we do it? • Step 1: Draw the line through C and T C T

  7. How? (continued) • Step 2: Draw a circle about T that passes through C, and let D denote the other end of that circle’s diameter C T D

  8. How? (continued) • Step 3: Construct the straight line which is the perpendicular bisector of segment CD tangent-line C T D

  9. Drawing a circle tangent to a line at a given point • At P, draw a perpendicular to the line AB • Set off the radius of the required circle on the perpendicular • Draw a circle with radius CP

  10. Drawing tangents to two circles • Move the triangle and T-square as a unit until one side of the triangle is tangent, by inspection, to the two cirles, • Then slide the triangle until the other side passes through the centre of one circle, and lightly mark the point of tangency, • Then slide the triangle until the side passes through the centre of the other circle, and mark the point of tangency. • Finally, slide the triangle back to the tangent lines between the two points of tangency. Draw the second tangent line in a similar manner

  11. Drawing two tangents circles Internally Tangent Externally Tangent

  12. How do we do it? External tangent • Circle 1: centre C and radius R1 • Circle 2: centreC’ and radius R2 • Step 1: Draw the line through C and T • Step 2: Set your compass for radius R2 • Step 3: Set your pointer on T. • Step 4: Make a mark on the right side of the line and label it C’ • Step 5: Set your pointer on C’ and draw the circle 2. T C C’

  13. How do we do it? Internal tangent • Circle 1: centre C and radius R1 • Circle 2: centreC’ and radius R2 • Step 1: Draw the line through C and T • Step 2: Set your compass for radius R2 • Step 3: Set your pointer on T. • Step 4: Make a mark on the left side of the line and label it C’ • Step 5: Set your pointer on C’ and draw the circle 2. T C C’

  14. Drawing an arc tangent to a line or arc and through a point

  15. Drawing an arc tangent to a line or arc and through a point

  16. Drawing an arc tangent to a line or arc and through a point

  17. C R1 R1 A B O D Drawing an Arc Tangent to a Line and an Arc • Given line AB and arc CD • Strike arcs R1 (given radius) • Draw construction arc parallel to given arc, with center O • Draw construction line parallel to given line AB • From intersection E, draw EO to get tangent point T1, and drop perpendicular to given line to get point of tangency T2 E T1 • Draw tangent arc R from T1 to T2with center E T2

  18. Drawing an arc tangent to an arc and a straight line

  19. A D R R R B C E Drawing an arc tangent to two lines at Right Angles • Given two lines AB and BC with right angle ABC • With B as the point, strike arc Requal to given radius O • With D and E as the points, strike arcs Requal to given radius • With O as the point, strike arc R equal to given radius

  20. Drawing an arc tangent to two lines at Right Angles

  21. A R B R C D Drawing an Arc Tangent to Two Lines at an Acute Angle • Given lines AB and CD • Draw parallellines at distance R • Draw the perpendiculars to locate points of tangency O • With O as the point, construct the tangent arc using distance R

  22. Drawing an arc tangent to two lines at acute or obtuse Angles

  23. A R C B R D Drawing an Arc Tangent to Two Lines at an Obtuse Angle • Given lines AB and CD • Construct parallel lines at distance R • Construct the perpendiculars to locate points of tangency O • With O as the point, construct the tangent arc using distance R

  24. Drawing an Arc Tangent to Two Lines at an Obtuse Angle

  25. Drawing an Arc Tangent to Two Arcs A R1 R O B C R1 D S • Given arc AB with center O and arc CD with center S • Strike arcs R1 = radius R • Draw construction arcs parallel to given arcs, using centers O and S E T • Join E to O and E to S to get tangent points T T • Draw tangent arc R from T to T, with center E

  26. Drawing an Arc Tangent to Two Arcs

  27. Drawing an arc tangent to two arcs enclosing one or both

  28. Tangents to Curves

  29. Tangent to an ellipse • Given an ellipse, and any point on it, we can draw a straight line through the point that will be tangent to this ellipse F1 F2

  30. How do we do it? • Step 1: Draw a line through the point T and through one of the two foci, say F1 T F1 F2

  31. How? (continued) • Step 2: Draw a circle about T that passes through F2, and let D denote the other end of that circle’s diameter T D F1 F2

  32. How? (continued) • Step 3: Locate the midpoint M of the line-segment joining F2 and D T D M F1 F2

  33. How? (continued) • Step 4: Construct the line through M and T (it will be the ellipse’s tangent-line at T) T D M F1 F2 tangent-line

  34. Proof that it’s a tangent • Observe that line MT is the perpendicular bisector of segment DF2 (because ΔTDF2 will be an isosceles triangle) T D M F1 F2 tangent-line

  35. Drawing circumcircle C Construct a Δ ABC Bisect the side AB Bisect the side BC o The two lines meet at O From O Join B B Taking OB as radius draw a circumcircle. A

  36. Bisect the BAC Bisect the ABC Taking O draw OP AB Drawing incircle C Construct a Δ ABC O The two lines meet at O Taking OP as radius A B P Draw a circumcircle

More Related