Competition · AMC preparation · step 4 of 4
AMC 8 · 2016 · #2
Grade 6 geometry-2d
Pick an answer.
AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
This is a 2D geometry problem with named points, so Tool #1 (Draw a Diagram) is the natural first move: sketch the rectangle, mark M as the midpoint of AD, and draw △ AMC. The picture immediately reveals a key fact — side AM lies along side AD of the rectangle, which is perpendicular to side DC. That means AM can serve as the base and DC gives the height for free. Tool #7 (Identify Subproblems) then splits the area calculation into two simple subproblems: (a) find the base AM, (b) find the perpendicular height, and (c) apply the triangle area formula.
Draw the rectangle
Sketch rectangle ABCD, mark M at the middle of side AD, and draw △ AMC — side AM sits on the rectangle's left edge.
Recognizing rectangle properties (opposite sides equal, adjacent sides perpendicular) from a sketched figure is a Grade 3 shape-attribute skill.
3.G.A.1Draw A DiagramFind the base
Subproblem 1 — the base: M halves AD = 8, so the base AM = 4.
Taking half of a whole number (the midpoint cuts a segment into two equal halves) is a Grade 4 fraction-of-a-whole operation.
4.NF.B.4Identify SubproblemsFind the height
Subproblem 2 — the height: DC ⊥ AD, so the height is DC = 6 (equal to AB).
Reading a perpendicular distance directly off a rectangle (right angle at D) uses Grade 4 understanding of perpendicular lines.
4.G.A.1Draw A DiagramApply the triangle area formula
Subproblem 3 — the area: ½ × base 4 × height 6 gives the area 12.
Using the triangle area formula with a known base and perpendicular height is the Grade 6 standard for finding triangle areas.
Triangle AMC's area is one half of base AM times height DC: 1/2 × 4 × 6 = 12.
▸ Why?
The area of a triangle is one half of a base times its matching perpendicular height.
▸ Why?
The base AM is 4, because M is the midpoint of AD, so AM and MD are equal parts that together make AD = 8.
▸ Why?
A whole segment is the sum of its parts, so if two equal parts make 8, each part is 4.
▸ Why?
The height is DC = 6, because DC is perpendicular to AD and, as the opposite side of the rectangle, has the same length as AB = 6.
▸ Why?
Opposite sides of a rectangle slide onto each other exactly, which is a rigid motion, so they have equal length.
This AMC 8 problem only needs the Grade 6 triangle area formula — pick a smart base where the height is already drawn for you, and the answer falls out in one step.
- Draw the rectangle
- Find the base
- Find the height
- Apply the triangle area formula
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