From a point P which is at a distance of 17 cm from the centre O of a circle of radius 8 cm, the pair of tangents PQ and PR to the circle are drawn. The area of triangle OPQ is :
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Step-by-Step Solution
Step 1: Identify given values and properties
We are given that the distance from point P to the center O is 17 cm, so OP=17 cm. The radius of the circle is 8 cm, so OQ=8 cm (where Q is the point of tangency). A key property of tangents is that the radius drawn to the point of tangency is perpendicular to the tangent. Therefore, △OPQ is a right-angled triangle with the right angle at Q.
Step 2: Calculate the length of the tangent PQ
Since △OPQ is a right-angled triangle at Q, we can use the Pythagorean theorem to find the length of the tangent PQ. The hypotenuse is OP, and OQ is one of the legs. Substituting the given values, we calculate PQ.
Step 3: Calculate the area of triangle OPQ
The area of a right-angled triangle is half the product of its perpendicular sides. In △OPQ, OQ and PQ are the perpendicular sides. We substitute their lengths to find the area.