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PQ is a tangent to a circle with centre O at the point P. If OPQ is an isosceles triangle with P as a vertex , then OQP is equal to
a) 30
b) 45
c) 60
d) 90

Answer
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Hint: We are given that PQ is a tangent to a circle with centre O at the point P. Let us consider OP to be the radius of the circle. Then, as PQ is tangent at point P, we know the tangent makes a right angle with the radius. Hence, OPQ=90. Also, we are given that OPQ is an isosceles triangle. We will understand this using a diagram.
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In the given figure, OP is the radius of the circle and PQ is tangent at point P. Hence, OPQ=90. Since OPQ is an isosceles triangle, we have two sides of this triangle are equal. Let us consider OP=PQ. Now, we will use the property of an isosceles triangle that angles opposite to equal sides are equal. After that we will use the angle sum property of a triangle i.e. Sum of angles in a triangle is equal to 180 and find the required angle.

Complete step-by-step solution:
Since PQ is a tangent at point P and OP is the radius of the circle.
We see it through a figure.
seo images

We know, tangent makes a right angle with the radius and So,
OPQ=90(1)
As OPQ is an isosceles triangle with OP=PQ.
We know, Angles opposite to equal sides are equal.
Angle opposite to PQ is QOP.
Angle opposite to OP is OQP.
Hence, OQP=QOP(2)
In OPQ, using Angle Sum Property
OPQ+OQP+QOP=180(3)
Using (1) and (2) in (3)
90+OQP+OQP=180
90+2OQP=180
Subtracting 90 both the sides, we get
90+2OQP90=18090
Clubbing the like terms on left hand side
2OQP+(9090)=18090
Solving the left and the right hand side
2OQP+0=90
2OQP=90
Dividing both the sides by 2.
2OQP2=902
OQP=45
Hence, we got OQP=45.
Therefore, the correct option is (b).

Note: We need to be very thorough with the properties of circles and triangles. Once we interpret the given question, we need to think of the steps of solving the problem. While solving, we need to take care of the calculations. We have to read the question carefully and then draw the figure and then apply the properties. Each and every detail in the question needs to be read very carefully. While solving, we have to take care of each and every step.