#### Question

In the figure, given below, AC is a transverse common tangent to two circles with centres P and Q and of radii 6 cm and 3 cm respectively.

Given that AB = 8 cm, calculate PQ.

#### Solution

Since AC is tangent to the circle with center P at point A.

∴ `∠`PAB =90°

Similarly, `∠`QCB = 90°

In ΔPAB and ΔQCB

`∠`PAB = `∠`OCB = 90°

`∠`PBA = `∠`QBC (vertically opposite angles)

∴ Δ PAB ˜ ΔQCB

⇒ `"PA "/"QC" = "PB"/"QB" ……… (i)

Also in Rt. ΔPAB,

`PB = sqrt(PA^2 + PB^2)`

⇒` PB = sqrt(6^2 + 8^2) = sqrt(36 + 64 ) = sqrt(100) = 10 cm`…….(ii)

From (i) and (ii)

`6/3 = (10 )/(QB)`

⇒ `QB = (3 × 10 )/6 = 5 cm`

Now,

`PQ = PB + QB = (10+5) cm = 15cm

Is there an error in this question or solution?

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In the Figure, Given Below, Ac is a Transverse Common Tangent to Two Circles with Centres P and Q and of Radii 6 Cm and 3 Cm Respectively. Given that Ab = 8 Cm, Calculate Pq. Concept: Tangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers.

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