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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.

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Tangents

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  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. Drawing an Arc Tangent to a Line and an Arc C R1 R1 A B O D • 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. Drawing an arc tangent to two lines atRight Angles A D R R R B C E • 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

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