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Question

In the diagram, PQ is a tangent to the circle with center O, at P. QRS is a straight line. Find the value of $$x$$.
95374_7cd645349d824d5c9e9c192d5a6ab8d3.png


A
25
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B
35
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C
45
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D
75
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Solution

The correct option is C $$35$$
$$\angle SRP=\dfrac {150}{2}=75^0$$     (angle at centre is twice the angle at the circumference)
$$\therefore \angle PRQ=180-75=105^0$$
Join OR,
$$\angle ORS=25^0$$
[since $$OS=OR=Radius]$$
$$\angle ORP=75-25=50^0$$
$$\angle OPR=\angle ORP=50^0$$
$$[OR=OP=Radius]$$
$$\angle RPQ=90-50=40^0 [OP\perp PQ]$$
$$\angle RPQ+\angle PRQ+\angle RQP=180^0$$
$$\angle RQP=180-[40+105]$$
$$=180-145$$
$$\therefore \angle RQP=35^0$$

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