We have , $$ ext{Hypotenuse}^{2}$$ = $$ ext {base}^{2}$$ + $$ ext {perpendicular}^{2}$$
Radii of the circles which intersect the tangents are parallel as both of them are perpendicular to the tangent.
Now, we draw a line parallel to the line which joins the centre of both the circles which intersects the extended radius of small circle at A and let the extended length be ‘a’
So, R = r + a i.e a = R - r
Now a right angled triangle is formed as shown in the figure as tangents and radii intersect at 90°
Applying Pythagoras theorem:
$$ ext{(Length of tangent)}^{2}$$ + $$a^{2} = d^{2}$$
$$ ext{(Length of tangent)}^{2}$$ = $$d^{2} - (R - r)^{2}$$
Length of tangent = $$sqrt{d^{2} - (R - r)^{2}}$$