AMC 10 · 2016 · #15
Grade 8 geometry-2dPick an answer.
The whole problem is about positions of three centers above a line, so tool #1 (Draw a Diagram) earns its keep: put l on the x-axis, and each center lands directly above its tangent point at a height equal to its radius. Tool #4 (Introduce a Variable) names the unknown horizontal positions of the centers and pins them down, because external tangency turns each center-to-center distance into a known number, and a right triangle (Pythagorean theorem) converts that into a horizontal gap. Once all three centers have coordinates, tool #7 (Identify Subproblems) computes the triangle's area cleanly by dropping the centers to the line and subtracting trapezoid areas, avoiding any messy formula.
Set up a coordinate frame
Each centre's height is its own radius.
A circle resting on a line balances directly above the point it touches, one radius high.
6.G.A.3Draw A DiagramTurn tangency into distances
Tangency gives both centre distances.
When two circles just touch from outside, the straight path between centers passes through the touch point, so its length is the radii added.
When two circles just touch from outside, the straight path between centres is the two radii added.
▸ Why?
The touching point lies on the line joining the centres, so that line is the two radii end to end.
▸ Why?
Every point of a circle sits one radius from its centre, so those two radii are the only lengths involved.
Find the horizontal gaps
The horizontal gaps follow by Pythagoras.
Knowing the slanted center distance and the height drop, the leftover flat distance pops out of the Pythagorean theorem.
8.G.B.7Introduce A VariableWrite the three centers' coordinates
That places all three centres.
With one tangent point pinned at zero, the two gaps slide the other centers into place.
6.NS.C.8Draw A DiagramGet the area by subtracting trapezoids
Subtracting trapezoids gives choice (D).
The triangle is just the wide trapezoid with the two narrow ones carved out.
6.G.A.1Identify SubproblemsStand each circle's center straight above where it touches the line at a height equal to its radius, use the Pythagorean theorem for the sideways gaps, then carve the triangle out of trapezoids to get √(6)-√(2).
- Set up a coordinate frame
- Turn tangency into distances
- Find the horizontal gaps
- Write the three centers' coordinates
- Get the area by subtracting trapezoids