Depth · one concept from first principles · step 2 of 4
The largest circle's diameter is 40 cm. A radius is half the diameter, so the largest radius is 20 cm.
Diameter is twice the radius, so halving the diameter gives the radius.
3.G.A.1Identify SubproblemsThe medium circle's diameter is the largest circle's radius, 20 cm. Half of that is the medium radius: 10 cm.
The medium circle stretches from the shared point to the largest circle's center, so its diameter is exactly one largest radius.
3.G.A.1Draw A DiagramThe medium circle's diameter is the same length as the largest circle's radius.
Both lengths are measured by one and the same piece of the line: the piece from the shared point at the left to the center of the largest circle.
That piece runs from a point on the largest circle to the largest circle's center, so it is one radius of the largest circle, the same length as every other radius of it.
The medium circle meets the line at the shared point and at the largest circle's center, and its own center sits on the line between them, so that piece is two medium radii joined end to end, which is the medium circle's diameter.
The medium diameter equals that piece and that piece equals the largest radius, so the medium diameter equals the largest radius.
The smallest circle's diameter is the medium circle's radius, 10 cm. Half of that is the smallest radius: 5 cm.
The smallest circle stretches from the shared point to the medium circle's center, so its diameter is one medium radius; halving it is a division within 100.
3.OA.C.7Draw A DiagramWhen one circle's diameter is the next circle's radius, every step inward is one more halving: 40, 20, 10, 5!
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Path Length Comparison & Shortest Path. Why the shapes repeat →