HKDSE Mathematics RONALD HUI TAK SUN SECONDARY SCHOOL

8 October 2015RONALD HUI

Book 5A Chapter 2 Tangents from an External Point

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

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

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

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.

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

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 

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 △

8 October 2015RONALD HUI