The illustration shows two circles of radii 4 cm and 2 cm respectively. the distance between the two centre is 8 cm. find the length of the common tangent.

The picture in the attached figure

we know that
ZY=4 cm
AB=XY------> 2 cm
ZX=ZY-XY------> 4-2----> 2 cm

in the right triangle AXZ
applying the Pythagorean Theorem
AZ²=AX²+XZ²-------> solve for AX
AX²=AZ²-XZ²------> 8²-2²-----> AX²=60------> AX=√60-----> AX=2√15 cm
BY=AX
BY=2√15 cm-------> the length of the common tangent

the answer is
the length of the common tangent is 2√15 cm

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