The centres of 2 circles with radii 8 cm and 3 cm are 13 cm apart A direct common - Maths - Circles

Question

The centres of 2 circles with radii 8 cm and 3 cm are 13 cm apart A
direct common tangent touches the circle at A and B respectively
What is the length of AB ?

Ashwini Kumar · Accepted Answer

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We have,OA is radius and AB is tangent, therefore ∠OAB=90°PB is radius and AB is tangent, therefore ∠PBA=90°PB=8 cmOA=3 cmOP=13 cmDraw OM⊥PB
∴MP=8-3=5 cmNow, △OMP is right angled triangle
∴OM2+MP2=OP2⇒OM2=OP2-MP2⇒OM2=132-52⇒OM2=169-25⇒OM2=144⇒OM=12 cmNow, ABMO is a rectangle
∴AB=OM=12 cmHence, AB=12 cm

The centres of 2 circles with radii 8 cm and 3 cm are 13 cm apart .A direct common tangent touches the circle at A and B respectively .What is the length of AB ?