Question

circle O has a diameter D and circle P has a diameter d. the two circles are externally tangent. find the length of the common external tangent in terms of the circles' diameters

Answer 1

let D/2 = x, and d/2 = y , where D > d

make a sketch, draw in the common tangent, join OP, and draw a line parallel to the tangent from P.
I see a rectangle and a right-angled triangle. The hypotenuse H of the right-angled triangle is equal to the length of the common tangent.
H^2 = (x+y)^2 + (x-y)^2
= x^2 + 2xy + y^2 + x^2 - 2xy + y^2 = 2x^2 + 2y^2
= 2(D^2/4) + 2(d^2/4)
= (1/2)(d^2 + D^2)

H = common tangent = (1/√2)√(D^2 + d^2)