Suppose x is any positive number. Circle 1: center (0, 0) and radius 2x

Circle 2: center (0, 0) and radius 10x

Why is circle 1 similar to circle 2?

Step 1:  Determine why circle 1 is similar to circle 2

The reason why circle 1 is similar to circle 2 is that they both expand at the same rate.  While x grows larger, both circle 1 and circle 2 grow at a proportional rate.  Circle 2 grows is 5 times bigger than circle 1 given any value 5.

When x = 3:  circle 1 = 6 and circle 2 = 30

When x = 10:  circle 1 = 20 and circle 2 = 100

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4