AMC 10 · 2013 · #13
Grade 8 geometry-2dPick an answer.
Nothing here is numeric. The problem is three overlapping structural conditions — a four-term progression, a fixed similarity, a three-term progression — plus a maximum to find, so the first move is to name things well. Two namings carry the whole solution. First, the similarity collapses every angle in the picture down to three values α, β, γ. Second, the quadrilateral's progression is written around its centre instead of its first term: four terms m-3e, m-e, m+e, m+3e. That second choice is the one that pays, because the sum condition then fixes m = 90 with no fraction in sight, the quantity being maximized becomes a plain multiple of the spread e, and the question turns into "how wide can the spread be". Once both namings are in place the conditions reduce to a short closed list of cases, so a systematic list finishes the search; and because a maximum is claimed, the winning case must be drawn, not just solved for.
Collapse the figure to three angles
Similarity collapses the figure to three angles.
Similar triangles have equal corresponding angles, so the entire picture runs on just three angle values.
Similar triangles have equal corresponding angles, so the entire picture runs on just three angle values.
▸ Why?
Triangles of the same shape match angle for angle, so no new angle can appear anywhere.
▸ Why?
Each triangle's three angles add to a straight angle, so knowing two of them names the third.
Centre the progression at ninety
The quadrilateral's steps centre on ninety.
Centering a progression at its average removes the unknown starting term and turns "largest sum" into "widest spread".
8.EE.C.7Introduce A VariableRead the corners off the split
The split names each corner directly.
The diagonal leaves two corners untouched and adds the same angle α to the other two.
7.G.B.5Draw A DiagramTwo pairs, one shared gap
Two pairs share the same gap.
The same angle α is added twice, so the progression has to contain two equal gaps — and four terms can only be paired up three ways.
8.EE.C.8Make A Systematic ListThe triangle condition means sixty
The triangle condition forces sixty.
Three terms in progression average to their middle one, so a fixed total of 180° nails that middle term at 60°.
8.G.A.5Introduce A VariableSix equations, two survivors
Only two step sizes survive.
Only a handful of equations can possibly hold, so test each one and keep the survivor with the widest spread.
8.EE.C.7Extreme PrincipleBuild the winner, do not assume it
Building the winner gives 240, choice (D).
Two rays whose angles with a segment total less than a straight angle always meet, so the figure can be drawn rather than merely hoped for.
7.G.A.2Draw A DiagramThe diagonal adds the same angle to two of the four corners, so the four corner angles must split into two pairs with the same gap — and that one structural fact decides the whole progression.
- Collapse the figure to three angles
- Centre the progression at ninety
- Read the corners off the split
- Two pairs, one shared gap
- The triangle condition means sixty
- Six equations, two survivors
- Build the winner, do not assume it