AMC 10 · 2009 · #9
Grade 8 geometry-2dPick an answer.
Plot what is fixed and what is free. The two fixed vertices sit on the line x+y=3, and C is confined to x+y=7 — the same slope. That parallel pair is the entire problem: it locks the height above base AB before C is ever chosen. So the order of work is (1) prove the height cannot change, then (2) measure the base and that one height. Picking a convenient C first would produce a number but would not show the number is forced.
Put base AB on a line
The two fixed points lie on a line parallel to the other one.
Two points fix a line, and matching forms makes the comparison instant.
8.EE.B.6Draw A DiagramParallel lines freeze the height
Parallel lines freeze the height.
Sliding a point along a line parallel to the base moves it sideways, never nearer or farther.
Sliding a point along a line parallel to the base moves it sideways, never nearer or farther.
▸ Why?
Two parallel lines keep a constant gap everywhere along their length.
▸ Why?
Triangles on one base between the same parallels therefore all have the same area.
Measure the base
The base measures 3√(2).
A run of 3 and a rise of 3 give the diagonal of a 3-by-3 square.
8.G.B.8Identify SubproblemsMeasure the gap between the lines
The gap between the lines measures 2√(2).
In an isosceles triangle, the line from the apex to the midpoint of the base hits it at a right angle.
8.G.B.7Identify SubproblemsHalf of base times height
Half of base times height gives 6, choice (C).
Half of base times height, with both factors now locked in.
7.G.B.6Identify SubproblemsWhen the free point can only slide along a line parallel to the base, the height is already decided — so the area is locked in before you even pick the point.
- Put base AB on a line
- Parallel lines freeze the height
- Measure the base
- Measure the gap between the lines
- Half of base times height