AMC 10 · 2017 · #24
Grade 11 geometry-2dPick an answer.
The question asks for a ratio, so the picture has no fixed size — only a shape. Name the two lengths that control that shape, (AB = a) and (BC = b), and every other length in the figure is forced by the two similarity statements. Each similarity is a free equation: the first one pins down (CD), the second one pins down where (E) is. Once (E) has coordinates, the point splits the quadrilateral into four triangles, so the area condition becomes one algebraic equation in (a) and (b). Dividing through by (b⁴) leaves a single equation in (a/b), which is exactly what was asked for.
Two right angles force a trapezoid
The two right angles force a trapezoid.
Two sides that both stand square on (BC) must be parallel, so the shape is a trapezoid with (BC) as its height.
Two sides that both stand square on the same segment must be parallel, so the shape is a trapezoid.
▸ Why?
A line crossing two lines at equal angles makes them parallel, and a right angle at each is such a pair.
▸ Why?
Parallel lines keep a constant gap everywhere, so the shared segment serves as the trapezoid's height.
Name the sides, then find CD
The first similarity fixes the remaining side.
The similarity says (BC) is the geometric mean of the two parallel sides, so (CD) is not free — it is decided by (a) and (b).
10.G-SRT.B.5Introduce A VariableRead the second similarity carefully
The second similarity must be read in vertex order.
Reading the vertex letters in order tells you exactly which side copies which, so one scale factor delivers both legs at once.
10.G-SRT.A.2Introduce A VariablePin E down with coordinates
Coordinates pin down the point E.
A right angle at (E) traps (E) on the circle with diameter (BC), and the altitude relations convert its two leg lengths into exact coordinates.
10.G-SRT.C.8Identify SubproblemsThree triangles share one area
Three small triangles have equal areas.
Base and height trade off perfectly on the three outer triangles, so they all have exactly the same area no matter what (a) and (b) are.
10.G-GPE.B.7Identify SubproblemsTurn the 17 into an equation
The seventeen-times condition becomes one equation.
Because three pieces are identical, "17 times" upgrades to "the whole trapezoid is 20 small triangles", which is a single equation.
9.A-CED.A.1Introduce A VariableReduce to the ratio alone
Tidying leaves a quartic in the ratio alone.
Dividing by (b⁴) throws away the size of the drawing and keeps only its shape, which is all the question ever asked about.
9.A-REI.B.4Introduce A VariableChoose the root and unnest the radical
Unnesting the radical gives two plus root five.
The two roots are reciprocals of each other, so the condition (AB ＞ BC) alone decides which one is the real answer.
11.N-RN.A.2Eliminate PossibilitiesEvery similarity written in a problem is a free equation — name the two lengths that control the shape, cash in each similarity, and the geometry collapses into one equation you can solve.
- Two right angles force a trapezoid
- Name the sides, then find CD
- Read the second similarity carefully
- Pin E down with coordinates
- Three triangles share one area
- Turn the 17 into an equation
- Reduce to the ratio alone
- Choose the root and unnest the radical
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