The figure above shows five congruent circles each with radius 2 such
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27 Oct 2022, 20:04
From the center of each circle to the point of Tangency, a rectangle is formed in 5 places. The rectangle has length of X, which is = 2 of the radii or 4.
If instead of following the Arcs of length Y, you connect each end point to each endpoint with a chord, you end up with a 10 sided figure ——- the 10 sided figure is created by connecting all the 5 X lengths with all the 5 chords of length Y.
We need to determine the central angle subtended by the ARC Y.
The sum of the interior angles of the 10 sided figure = (10 - 2) (180) = 1440 degrees
At each vertex where the radius meets the point of Tangency, a 90 degrees angle is formed. We have 10 such angles.
1440 - (10) (90) = 540
This leaves 540 degrees to be placed evenly at the 10 points at each vertex.
Or, 54 degrees at each vertex next to the point of Tangency on each circle.
Looking at one triangle formed within a circle: the base angles across from the 2 radii are 54.
Therefore, the central angle subtended by ARC Y is:
180 - (2) (54) = 72
We have 5 ARCS, so:
(5) (72 / 360) (2 * 2 * (pi)) = the 5 circular Y-ARCS =
4(pi)
And
The length of EACH of the 5 X chords, from point of Tangency to point of Tangency = 2(radius) = 2(2) = 4
5X = 5(4) = 20
Answer: perimeter = 5X + 5Y =
4(pi) + 20
*B*
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