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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. - ICSE Class 10 - Mathematics

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ConceptTangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers

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

 

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Solution 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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