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HKDSE Mathematics RONALD HUI TAK SUN SECONDARY SCHOOL.

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Presentation on theme: "HKDSE Mathematics RONALD HUI TAK SUN SECONDARY SCHOOL."— Presentation transcript:

1 HKDSE Mathematics RONALD HUI TAK SUN SECONDARY SCHOOL

2 8 October 2015RONALD HUI

3 Book 5A Chapter 2 Tangents from an External Point

4 For any point outside the circle, A tangent we can always draw two tangents to the circle from the point. B

5 B tangent For any point outside the circle, we can always draw two tangents to the circle from the point.

6 Do the two tangents TP and TQ have any properties? Yes, there are three properties. T P Q O

7 T P Q O Consider △ OPT and △ OQT. OP = OQradii OT = OT common side ∠ OPT = ∠ OQT = 90  tangent ⊥ radius ∴ △ OPT △ OQT RHSRHS Hence, (i) TP = TQ corr. sides, △ s (ii) ∠ POT = ∠ QOT corr. ∠ s, △ s (iii) ∠ PTO = ∠ QTO corr. ∠ s, △ s Join OP, OQ and OT.

8 Theorem 2.4 If two tangents, TP and TQ, are drawn to a circle from an external point T and touch the circle at P and Q respectively, then Abbreviation: tangent properties (i) TP = TQ (ii) ∠ POT = ∠ QOT (iii) ∠ PTO = ∠ QTO T P Q O

9 In the figure, TP and TQ are tangents to the circle at P and Q respectively. Find x. ∵  TP = TQ tangent properties P T Q 55  x base  s, isos. △ ∴ x = 55 

10 Follow-up question In the figure, TA and TB are tangents to the circle at A and B respectively. TOC is a straight line. Find x. A O T B C 27  x  OTA =  OTB  OAT = 90  In △ OAT, 27  = 27  = 90  + 27  x = ∠ OAT + ∠ OTA = tangent  radius tangent properties ext. ∠ of △

11 8 October 2015RONALD HUI


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