Competition · AMC preparation · step 4 of 4
AMC 8 · 2005 · #4
Grade 5 geometry-2dPick an answer.
AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The question chains three small tasks behind one short sentence, so Tool #7 (Break into Subproblems) keeps the work organized: (i) add the three triangle sides to get the shared perimeter, (ii) divide by 4 to get the square's side length, (iii) square that side to get the area. Tool #1 (Draw a Diagram) is the supporting move — a quick sketch of the triangle and square reminds us that "equal perimeter" is the only link between the two shapes, and that the square's four equal sides give the ÷ 4 step.
Add the three sides
Subproblem 1: add the triangle's three sides to get its perimeter.
Adding decimals to the tenths place is Grade 5 arithmetic — line up the decimal points and add column by column.
5.NBT.B.7Identify SubproblemsDivide the perimeter by 4
Subproblem 2: the square shares that 24 cm perimeter, so divide by 4 for one equal side.
The Grade 3 perimeter formula for a square is P = 4s, so s = P/4. Equal perimeters means the 24 carries straight from the triangle to the square.
The square's side length must be 6 cm.
▸ Why?
The square's whole boundary measures 24 cm, and its four equal sides share that boundary evenly, so one side is 24 ÷ 4 = 6 cm.
▸ Why?
The square's boundary is 24 cm because it is given to equal the triangle's perimeter, and going all the way around the triangle covers 6.1 + 8.2 + 9.7 = 24 cm.
▸ Why?
A shape's perimeter is the full trip around its edge, so the triangle's perimeter is its three side lengths joined with no gaps and no overlaps.
▸ Why?
A square's four sides are all the same length, so the 24 cm boundary is four equal copies of one side, and recovering that side means undoing the times-four.
▸ Why?
Building the boundary multiplies one side by four, so sharing the boundary back into four equal parts by division reverses that step and returns the single side.
Square the side
Subproblem 3: square that 6 cm side to get the area.
Area of a square is side × side, a Grade 3 standard. A 6-by-6 square holds 36 unit squares.
3.MD.C.7Identify SubproblemsWhen two shapes share a perimeter, that one number is the only bridge between them — add the triangle's sides to cross the bridge, then the square's ÷ 4 and side-squared are routine Grade 3 work.
- Add the three sides
- Divide the perimeter by 4
- Square the side
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