AMC 10 · 2014 · #21
Grade 10 geometry-2d
Pick an answer.
The picture hands over no length at all, so nothing can be measured or chased directly. The real difficulty is that there are two hidden unknowns, not one: the width BE and the tilt of JKHG. Tool #4 (Introduce a Variable) names both of them up front — x=BE for the width, and a unit direction vector (u,v) for the tilt. Tool #1 (Draw a Diagram) and Tool #17 (Visualize Spatial Relationships) do the work of turning "JKHG is a rectangle" into coordinates: put the square on axes, then build the whole rectangle from J, the direction (u,v), and its quarter turn (-v,u), so being a rectangle costs no equations at all. Tool #13 (Convert to Algebra) then writes down the only facts the figure actually supplies — which corner is stuck to which line — as three plain equations. It is worth being strict here: a tempting shortcut is to stare at the picture, declare the four corner triangles similar and read off equal segments, but every one of those claims is a figure-reading, not a proof. The incidence conditions are the whole content, and they are enough. Tool #7 (Identify Subproblems) extracts the tilt u twice — once from the requirement that JG has length 1, once from the requirement that K lands on EF — and the two expressions must agree, which is the equation that decides everything. Tool #3 (Eliminate Possibilities) discards the two solutions that the algebra produces but the geometry forbids: the untilted case and the root bigger than 1.
Put the square on axes
Coordinates name every corner.
Coordinates turn "this corner sits on that side" into one plain equation each.
10.G-GPE.B.4Draw A DiagramName BE and cash in the congruence
The congruence transfers both dimensions.
Congruent rectangles are the same rectangle in a different pose, so both side lengths carry across unchanged.
8.G.A.2Introduce A VariableBuild the tilted rectangle from one direction
One direction builds the tilted rectangle.
One quarter turn of a direction vector builds the whole rectangle, so being a rectangle costs no equations.
One quarter turn of a direction builds the whole rectangle, so being a rectangle costs no extra equations.
▸ Why?
A quarter turn sends a direction to one at right angles, which is exactly what the next side needs.
▸ Why?
The turn moves the direction without stretching it, so the side lengths stay whatever they were chosen to be.
Write the incidence facts as equations
Each contact point is one equation.
The only real information in the figure is which corner is stuck to which line, and each of those is one equation.
9.A-CED.A.3Convert To AlgebraGet the tilt from the length condition
The length condition gives the tilt once.
Demanding that JG be exactly 1 unit long already pins the tilt in terms of x.
8.G.B.8Identify SubproblemsGet the tilt a second time from K
The contact condition gives it again.
The same tilt must also drop K onto EF, so a second expression for u appears.
9.A-SSE.A.2Identify SubproblemsMake the two tilts agree
Matching them gives a plain quadratic.
Two honest routes to the same tilt must give the same number, and that clash is the equation.
9.A-REI.B.4Eliminate PossibilitiesRebuild the figure and confirm
Rebuilding confirms 2-√3, choice (C).
Plugging the answer back in shows the picture really closes, so the configuration exists and is the one drawn.
10.G-GPE.B.4Draw A DiagramA tilted shape has two secrets — where it starts and how far it leans — so name both, write down which corner is forced onto which side, and the picture turns into one quadratic that answers itself.
- Put the square on axes
- Name BE and cash in the congruence
- Build the tilted rectangle from one direction
- Write the incidence facts as equations
- Get the tilt from the length condition
- Get the tilt a second time from K
- Make the two tilts agree
- Rebuild the figure and confirm