# The radii of two circles are 5cm and 2cm. the length of the direct common tangent of the circles is 1.5times the length of the transverse common tangent. what is the distance between the centres of the circles ?

1
by Anandhi06

2015-08-24T00:35:31+05:30

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Look at the diagram to derive the formulas.

Radii are :    R1 = 5 cm      R2 = 2 cm
C1 C2 is the distance between the centers.
lengths of direct tangent and transverse tangents are DT and TT respectively.
the formulas are :
DT² = C1 C2² - (R1 - R2)²
TT² = C1 C2² - (R1 + R2)²

Subtract one from the other:
DT² - TT² = (R1 + R2)² - (R1 - R2)²
= 4 R1 R2 = 40 cm²    --- (1)
DT = 1.5 TT
=>    (1.5² - 1) TT² = 40      =>  TT² = 32    => TT = 4√2 cm
=>  DT = 6 √2 cm

C1C2² = TT² + (R1 + R2)² = 32 + 49 = 81 cm²

C1 C2  = 9 cm

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