the radii of two circles are 4 and 7 while the distance between their centers is 13.Find the length of their common external tangent segment?

Since we can orient these circles in any fashion that we would like.....let the first circle be centered at (0,3)....and the top of this circle will be at (0,7)   [because its radius is 4]

Now....let the second circle also have its "top" at (x, 7), where x is the center of this circle....since the radius is 7, then this center will be on the x axis....[note that the tangent to both circles will just be a horizontal line passing through (0,3) and (x,7)

And since the center of the second circle is 13 units from the center of the first, we can solve this distance equation to find this center x coordinate

√ [ (x - 0)^2  + (3 -0)^2 ] = 13  →  √ [x ^2 + 9 ]  = 13  →  x^2 +  9 = 169  →  x^2 = 160  → x = 4√10

So.....the second circle will be centered at (4√10, 0 )

And the length of the common external tangent will just be "x"   = 4√10

See the situation, here : https://www.desmos.com/calculator/kpz4qhtdgl

the radii of two circles  are 4 and 7 while the distance between their centers is 13.Find the length of their common external tangent segment?

I like the similar triangle approach, Omi....!!!!!