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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.

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))

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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