In figure, CP and CQ are tangents to the circle with center O. ARB is another tangent touching the circle at R. If CP = 12 cm, and BC = 8 cm, then find the length of BR.
Hint: Read the question carefully, and consider all the given information as it leads to solution. Tangent to a circle is a line that intersects the circle at only one point. The property of Pair of tangents to a circle drawn from an external point is the key to solve this question.
Complete step by step solution: Step 1: Draw the labeled diagram carefully:
Figure 1: circle with center O Step 2: Given that: Line CP, CQ, and AB are tangents to the circle with center O. Length CP = 12 cm, BC = 8 cm Step 3: State the theorem related to Pair of tangents to a circle drawn from an external point: Theorem 1: The lengths of tangents drawn from an external point to a circle are equal.
Figure 2: tangents PQ and PR to the circle with center O Hence, length PQ = length PR Step 4: apply the theorem Here, line CP and CQ are tangents from external point C to the circle with center O. $ \Rightarrow $CP = CQ Hence, CQ = 12 cm Step 5: since, QBC is a straight line: Therefore, CQ = BQ + BC 12 = BQ + 8 BQ = 12 – 8 $ \Rightarrow $BQ = 4 cm. …… (1) Step 6: apply the theorem Here, line BQ and BR are tangents from external point B to the circle with center O. $ \Rightarrow $BQ = BR Hence, BR = 4 cm (from (1))
Therefore, the required length of the BR is 4 cm.
Note: Another important theorem of tangents and circle: Theorem: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Figure 3: tangent XY to the circle with center O. Hence, $OP \bot PQ$ There is only one tangent at a point of the circle.
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