AMC 10 · 2014 · #5
Grade 8 geometry-2d
Pick an answer.
The pane size is given only as a ratio 5:2, so Tool #4 (Introduce a Variable) names one scale factor x and turns the ratio into real lengths: width 2x, height 5x. Tool #1 (Draw a Diagram) reads the picture to count how many panes and borders lie along each side. Tool #7 (Identify Subproblems) splits the job into two totals — the width across and the height down — each built from panes plus borders. Setting those two totals equal (the window is square) gives one linear equation to solve for x.
Turn the ratio into lengths
The pane ratio becomes two lengths.
A ratio only fixes the shape, not the size; one scale factor unlocks the actual measurements.
6.RP.A.3Introduce A VariableAdd up the width across
Across, there are five borders.
Lining up n panes needs n+1 borders — one more border than there are panes.
Lining up several panes needs one more border than there are panes.
▸ Why?
Each pane sits between two borders and each inner border is shared, so the borders outnumber the panes by one.
▸ Why?
The panes and borders alternate at a fixed spacing, so the total width is a count of equal steps.
Add up the height down
Down, there are three.
Stacking 2 panes needs 3 borders, again one more border than panes.
6.EE.A.2Identify SubproblemsUse the square condition to solve
The square condition gives one equation.
Being a square forces width and height to match, which is exactly the equation that pins down x.
8.EE.C.7Introduce A VariableCompute the side length
Either direction gives 26, choice (A).
Both totals landing on the same number is the built-in proof that the answer is right.
6.EE.A.2Introduce A VariableName the pane size with one variable, add panes and borders to get the width and height, then set them equal because the window is a square.
- Turn the ratio into lengths
- Add up the width across
- Add up the height down
- Use the square condition to solve
- Compute the side length