AMC 10 · 2002 · #18
Grade 9 probabilityPick an answer.
Because the point is uniform, the probability is just an area ratio, so the real task is to draw the boundary between "nearer the origin" and "nearer (3,1)" and measure what it cuts off — that is Tool #1 (Draw a Diagram). To find that boundary exactly, Tool #4 (Introduce a Variable) names a generic point (x,y) and writes both distances, and Tool #13 (Convert to Algebra) turns the phrase "is closer to" into an inequality; comparing squared distances collapses that messy square-root comparison into a single straight-line inequality, so the boundary is a line and the two pieces are polygons whose areas are easy. Finally, the piece nearer (3,1) is a clean trapezoid while the piece nearer the origin is a five-sided blob, so Tool #16 (Change Focus / Count the Complement) says: measure the easy piece and subtract.
Name the point and both distances
Distances are never negative, so compare their squares and skip the roots.
Squaring is order-preserving on non-negative numbers, so a race between two distances is the same race between their squares.
8.G.B.8Introduce A VariableTurn the comparison into a line
The squared terms cancel, reducing the condition to the line 3x + y ＜ 5.
The squared terms are the same on both sides, so they cancel and what survives is linear — the divider between the two points is always a straight line.
The squared terms cancel, so the divider between the two points is a straight line.
▸ Why?
Points equally far from two fixed points lie on the mirror line between them, which crosses their joining segment square on.
▸ Why?
The point is scattered evenly over the rectangle, so the probability is just the share of area on the winning side.
Draw the divider across the rectangle
That line crosses the rectangle from (5/3, 0) to (4/3, 1), cutting off a trapezoid.
Finding where the divider meets the top and bottom edges pins down exactly how the rectangle is sliced.
6.G.A.3Draw A DiagramMeasure the easier piece
The far piece is a trapezoid of area 1/2, the easier of the two to measure.
The unwanted piece is a plain trapezoid, so measuring it and subtracting beats measuring the five-sided piece directly.
6.G.A.1Change Focus Count The ComplementConvert areas into the probability
So the near piece is 3/2 out of area 2, giving probability 3/4, choice (C).
With a uniformly random point, area is probability — nothing else about the shape matters.
7.SP.C.7Change Focus Count The ComplementThe points tied between two fixed points always form a straight line, so "who is closer" questions turn into cutting a region with a line and comparing areas.
- Name the point and both distances
- Turn the comparison into a line
- Draw the divider across the rectangle
- Measure the easier piece
- Convert areas into the probability
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Same archetype — closest grade level first.
