Competition · AMC preparation · step 4 of 4
AMC 8 · 2006 · #7
Grade 7 geometry-2dPick an answer.
AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The three circles are described in three different languages — radius, circumference, area. To compare them, Tool #12 (Change Representation) says: put them all in the same language. Radius is the natural choice because two of the three quantities are already one short formula away from it. Tool #15 (Use a Formula) then does the heavy lifting: C = 2π r gives Y's radius from its circumference, and A = π r² gives Z's radius from its area. Once all three radii are plain numbers, ordering them is just comparing π with 3 and 4.
Read circle X's radius
Circle X is already given as a radius — record r_X = π and move on.
Nothing to convert — X already speaks the radius language.
7.G.B.4Draw A Venn DiagramTurn circumference into radius
Convert Y's circumference with C = 2π r: 2π r_Y = 8π gives r_Y = 4.
The Grade 7 circumference formula is the bridge from C to r. Dividing by 2π is a one-step equation.
Circle Y is described only by its circumference 8π, yet that single fact forces its radius to be exactly 4.
▸ Why?
The circumference is the full distance once around the circle, and for every circle that distance is exactly 2π times the radius, so 8π must equal 2π · r_Y.
▸ Why?
Once 2π · r_Y is known to equal 8π, dividing both of these equal amounts by the shared factor 2π undoes the multiplication and leaves r_Y = 4.
Turn area into radius
Convert Z's area with A = π r²: π r_Z² = 9π gives r_Z² = 9, so r_Z = 3.
The Grade 7 area formula gives r² = 9, and the positive square root is 3.
7.G.B.4Organize Information In More WaysCompare the three radii
Compare the radii as numbers: since 3 ＜ π ＜ 4, we get r_Z ＜ r_X ＜ r_Y.
Once every circle is described by the same quantity, ordering them is just ordering three numbers on the number line.
7.NS.A.3Draw A Venn DiagramOrder them smallest to largest
Read the circles off in that radius order to get choice (B).
The ordering of the radii is the ordering of the circles.
7.G.B.4Organize Information In More WaysWhen circles are described in different ways — radius, circumference, area — convert every one into the same form first. Once all three are radii, ordering the circles is just ordering three numbers.
- Read circle X's radius
- Turn circumference into radius
- Turn area into radius
- Compare the three radii
- Order them smallest to largest
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