Question

# If PA and PB are tangents from an outside point P. such that PA =10 cm and $$\angle APB = 60$$. Find the length of chord AB.

Solution

## Consider a circle C with center O.We have PA and PB are tangents of the circle, PA=10cm and $$\angle APB=60^{o}$$Join OP,Such that,In $$\triangle PAC$$ and $$\triangle PBC$$ we have,$$PA=PB$$   [tangent of the circle fro the outer point p is equal]$$\angle PAC= \angle PBC$$   [angle made by the external tangent on a circle is equal]$$PC=CP$$  [common]so,$$\triangle PAC$$ $$\cong$$ $$\triangle PBC$$  [By SAS criteria]so,$$AC=BC$$ ........(i)$$\angle ACP= \angle BCP$$ ......(ii)since$$\angle APB= \angle APC+ \angle BPC$$so,$$\angle APC= \dfrac {1} {2} \times 60^0=30^0$$ $$\angle APC=30^0$$$$\angle ACP+ \angle BCP=180^0$$from equation 2 we get$$\angle ACP=\dfrac {1} {2} \times180^0=90^0$$Thus in Right $$\triangle$$ ACP$$sin$$$$30^0=\dfrac{AC}{AP}$$$$\dfrac{1}{2}=\dfrac{AC}{10}$$cm$$AC=5$$cmSince $$AC=BC$$so,$$AB= AC+BC$$     $$=5$$cm$$+5$$cm     $$=10$$cmMathematics

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