Semicircles On A Line

Relevant wiki: Circles - Circumference

Let the radius of the green semicircle be $R$ and the radius of the smaller (or if they are equal, then one of them) blue be $r$

Then:

• the radius of the bigger blue semicircle is $R - r$

• the length of the green line (the perimeter of the green semicircle) is: $R\pi$

• the perimeter of the smaller blue semicircle is: $r\pi$ .

• the perimeter of the bigger blue semicircle is: $(R - r)\pi$ .

• the length of the blue line: $r\pi + (R - r)\pi = r\pi + Rπ\pi- r\pi = R\pi$ .

Hence, our answer is:

$\boxed { \text {Equal Length } }$ .

Let D be the diameter of the big green semicircle, d be the diameter of the left blue semicircle, and δ be the diameter of the right blue semicircle.

IMPORTANT FACT 1: Notice that d + δ = D, since d and δ partition the line segment with length D.

Now the length of the blue line is the sum of the length of the blue semicircles:

π·d/2 + π·δ/2

= π·(d + δ)/2 (By the distriputive law)

= π·D/2 (By important fact 1)

But π·D/2 is the length of the green semicircle!

Thus the blue line and the green line have the same length.

Let a be the diameter of the larger blue semicircle and b be the diameter of the smaller blue semicircle. Taking the total perimeter of the blue semicircles, it would be (a pi)/2 + (b pi)/2. Analyzing the diameter of the green semicircle, we see it's a + b. Taking the diameter of the green semicircle, we see it's (a + b)*pi/2. Comparing our expressions, we see that they are the same. Therefore, the green and blue semicircle perimeters are equal to each other.