Competition · AMC preparation · step 4 of 4
AMC 8 · 2006 · #19
Grade 8 geometry-2d
Pick an answer.
AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #1 (Draw a Diagram) is the natural opening: redraw the figure and mark every piece of information — the two equal sides AB = BC, the two midpoint splits at D, and the known length CE = 11. Tool #12 (Use Symmetry) reads the congruence △ ABD ≅ △ ECD off the vertex order: the matching pairs are A ⇔ E, B ⇔ C, D ⇔ D, so AB matches EC. That single matching turns the given CE = 11 into AB = 11, and the isosceles condition plus the midpoint cuts BC exactly in half. No algebra is needed — three sentences of side-chasing finish it.
Mark up the figure
Redraw △ ABC with AB = BC, place E so D is the midpoint of both BC and AE, and label the one known length CE = 11.
Drawing and labeling each given is the Grade 4 "identify line segments" move that keeps the equal pieces visible.
4.G.A.1Draw A DiagramRead the congruence in order
The vertex order in △ ABD ≅ △ ECD pairs A–E, B–C, D–D, so side AB matches side EC — giving AB = 11.
Grade 8 congruence: corresponding parts of congruent triangles are equal. The vertex order is the dictionary that tells you which side equals which.
Side AB of the first triangle is exactly as long as side EC of the second.
▸ Why?
The congruence △ ABD ≅ △ ECD names the corners in matching order, pairing A with E and B with C, so the side from A to B corresponds to the side from E to C.
▸ Why?
Corresponding sides of congruent triangles have equal length, because two triangles are congruent exactly when one can be slid, turned, or flipped to rest perfectly on the other, and that motion carries each corner onto its partner without stretching or shrinking any side.
Carry the length across
The triangle is isosceles with AB = BC, and AB = 11, so BC = 11 too.
An isosceles triangle labels two sides as equal — once one side is known, the matching side is known too.
4.G.A.2Draw A DiagramHalve the side at the midpoint
D is the midpoint of BC, so BD is half of BC: BD = = 5.5.
A midpoint is the Grade 5 "half of a length" idea: cut 11 in half to get 5.5.
5.NF.B.4Draw A DiagramCongruent triangles match corner-to-corner in the order they are written. Once AB = EC = 11, the isosceles side BC is also 11, and the midpoint cuts it in half.
- Mark up the figure
- Read the congruence in order
- Carry the length across
- Halve the side at the midpoint
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