Competition · AMC preparation · step 4 of 4
AMC 8 · 2008 · #18
Grade 7 geometry-2d
Pick an answer.
AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The path is one curvy trip, but it is built from six clean pieces: three quarter-arcs and three straight segments. Tool #7 (Break the Problem into Subproblems) says to find the length of each piece by itself, then add. Tool #1 (Draw a Picture) helps us label each piece with its endpoints in coordinates so we know which circle the arc sits on and which radius the segment uses.
Break the path into pieces
Follow the arrows from A to K and split the trip into six pieces: three quarter-arcs and three straight segments.
Plotting the endpoints on a coordinate grid is the Grade 6 way of seeing the shape. Each piece is either an arc on a known circle or a segment along an axis.
6.G.A.3Draw A DiagramFind the big arc
The first piece is a quarter of the large circle (circumference 40π), giving an arc of 10π.
The Grade 7 circle formula C = 2π r gives the whole way around. A quarter turn uses a quarter of that.
The first piece of the path, the quarter-turn along the large circle from its top around to its side, has length 10π meters — one quarter of the whole way around that circle.
▸ Why?
The turn from the top of the large circle around to its side is a right angle at the center, one quarter of a full turn, so the piece is one quarter of the circle's loop; equal turns about the center cut off equal shares of the loop.
▸ Why?
Equal turns about the center sweep equal lengths of the circle, because the circle curves the same way all the way around — each of its points lies exactly one radius from the center.
▸ Why?
One full loop of the large circle is 2π times its radius, 2π(20)=40π meters, so one quarter of the loop is 1/4(40π)=10π meters.
▸ Why?
The whole way around a circle of radius r measures exactly 2π r, so the large circle with r=20 has a loop of 2π(20)=40π meters.
Find the two small arcs
Each small arc is a quarter of the small circle (circumference 20π), so each one is 5π.
Same circle formula, smaller radius. Two equal quarter-arcs add to 10π.
7.G.B.4Identify SubproblemsFind the straight segments
Each x-axis segment is 20-10=10; the y-axis segment is the small circle's diameter, 20.
Distance along an axis is just the difference in coordinates. Grade 6 number-line distance, used on the x- and y-axes.
6.NS.C.8Identify SubproblemsAdd all six pieces
Group π-terms and constants: 10π+5π+5π=20π and 10+20+10=40, so the total is 20π+40 → (E).
Combining like terms is a Grade 6 expression move: π terms stack together, constants stack together.
6.EE.A.3Identify SubproblemsA curvy path becomes easy when you cut it into pieces — each arc is a quarter of its circle and each straight piece is a difference of coordinates.
- Break the path into pieces
- Find the big arc
- Find the two small arcs
- Find the straight segments
- Add all six pieces
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