Referring to the attached pic:

(1) prove triangle BTQ ~ triangle QTA

(2) find TQ

(3) find the values of the radii

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- Nov 1st 2009, 08:30 AMukorovquestion on properties of circle 3
Referring to the attached pic:

(1) prove triangle BTQ ~ triangle QTA

(2) find TQ

(3) find the values of the radii - Nov 2nd 2009, 07:55 AMGrandad
Hello ukorovProduce $\displaystyle QO_2$ to meet the circle at $\displaystyle R$. Then $\displaystyle \angle TBQ = \angle BRQ$ (alternate segment)

$\displaystyle \Rightarrow \angle TBQ = \tfrac12 \angle BO_2Q$ (angle at centre)

$\displaystyle \angle TAO_1 = \angle TBO_2 = 90^o$ (angle between tangent and radius)

$\displaystyle \Rightarrow AO_1 \parallel BO_2$ (corresponding angles equal)

$\displaystyle \Rightarrow \angle AO_1T = \angle BO_2Q$ (corresponding angles)

But $\displaystyle \angle AQT =\tfrac12\angle AO_1T$ (angle at centre)

$\displaystyle \Rightarrow \angle AQT = \tfrac12\angle BO_2Q =\angle TBQ$ (proved)

So in $\displaystyle \triangle$'s $\displaystyle BTQ, QTA$:$\displaystyle \angle AQT = \angle TBQ$ (proved)$\displaystyle \Rightarrow \triangle BTQ \sim \triangle QTA$

$\displaystyle \angle ATQ = \angle QTB$ (same angle)

Grandad - Nov 2nd 2009, 09:22 AMukorov
- Nov 2nd 2009, 10:42 AMGrandad
- Nov 2nd 2009, 06:24 PMukorov