Competition · AMC preparation · step 4 of 4
AMC 8 · 2001 · #8
Grade 5 geometry-2drate-ratio
Pick an answer.
AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The figure is on a one-inch grid, so Tool #1 (Draw a Diagram) is built in — read each diagonal straight off the grid by counting units between opposite vertices. Tool #9 (Solve an Easier Related Problem) handles the scaling: get the answer for the easy small kite first, then multiply by 3 for the large kite. Together these two tools avoid any need for algebra or distance formulas.
Read the small kite's diagonals
Read the small kite's two diagonals off the grid by counting units: the vertical is 7 in and the horizontal is 6 in.
Both diagonals run along grid lines, so reading their length is just counting unit dots between the endpoints — exactly the Grade 5 coordinate-plane skill.
5.G.A.1Draw A DiagramScale up by 3
Tripling the grid multiplies every length by 3, so the diagonals become 21 in and 18 in.
Solving the small (easier) kite first turns the large kite into a one-step scaling: multiply each known diagonal by the factor 3.
Each diagonal of the large kite is 3 times as long as the matching diagonal of the small kite, so the 7-inch and 6-inch diagonals become 3 × 7 and 3 × 6 inches.
▸ Why?
Tripling the grid multiplies every point's coordinates by 3, and each of these two diagonals runs straight along the grid, so its length is simply the difference between two of those coordinates.
▸ Why?
Tripling the far and near coordinates and then subtracting gives the same result as subtracting first and then tripling: 3b - 3a = 3(b - a), so a diagonal of length 7 becomes 3 × 7 and one of length 6 becomes 3 × 6.
▸ Why?
For a segment lying along a grid line, the run from the origin to the near end plus the run from the near end to the far end make up the whole run to the far end, so the segment's length is the far coordinate minus the near coordinate.
Add the two diagonals
The cross is the two large diagonals, so add them: 21 + 18 = 39 in of bracing.
A cross made of two diagonals uses one piece for each diagonal, so adding the two diagonal lengths gives the total material — a one-step Grade 4 word-problem sum.
4.OA.A.3Draw A DiagramOn a grid, a diagonal that runs straight across rows or columns is just a counting job — 7 up, 6 across. Triple the grid and you triple each length (21 and 18), then add the two diagonals for the cross-shaped brace: 21 + 18 = 39 inches, choice (E).
- Read the small kite's diagonals
- Scale up by 3
- Add the two diagonals
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