AMC 10 · 2005 · #18
Grade 10 geometry-2dPick an answer.
"All three angles acute" is three conditions at once, so we split it into three subproblems (Tool #7), one per vertex. Naming C = (x,y) (Tool #4) turns each condition into an inequality, because an angle is acute exactly when the dot product of the two vectors leaving that vertex is positive. Each inequality has a picture — two half-planes and the outside of a circle — so drawing the diagram (Tool #1) shows R is a diagonal strip with a disk punched out. The area is then easiest as a complement: take the strip and subtract the disk (Tool #16). One check is essential and easy to forget: the disk must lie entirely inside the strip, or subtracting all of it would remove area that was never there.
Test each angle with a dot product
Each angle becomes a dot product test, handled one at a time.
A dot product is positive, zero, or negative exactly as the angle is acute, right, or obtuse.
10.G-GPE.B.4Identify SubproblemsAngle A cuts a half-plane
The first fixed point cuts a half-plane.
Sliding C across the perpendicular at A is exactly when the angle at A swings past a right angle.
10.G-GPE.B.4Introduce A VariableAngle B cuts the other half-plane
The second cuts the other, leaving a diagonal strip.
The same perpendicular test at the far endpoint supplies the strip's second edge.
10.G-GPE.B.4Introduce A VariableAngle C excludes a disk
The third completes the square into a condition about a circle.
Completing the square is what turns a quadratic inequality into "far enough from this centre".
10.G-GPE.A.1Introduce A VariableRecognise the circle on AB
That circle has the fixed segment as its diameter.
Every point on the circle with diameter AB sees that segment as a right angle, so the disk is exactly the obtuse zone.
Every point on the circle with AB as diameter sees that segment at a right angle, so the disk is exactly the obtuse zone.
▸ Why?
All points of that circle are the same distance from the midpoint of AB, which is half the segment.
▸ Why?
An angle is right exactly when the squares on the two sides add up to the square on AB, and inside the disk that sum falls short.
Check the disk fits inside the strip
Checking tangency shows the disk sits wholly inside the strip.
The two perpendiculars touch the circle at A and B and nowhere else, so no part of the disk pokes outside the strip.
10.G-GPE.B.4Draw A DiagramArea of the strip
Two triangles give the strip area 90.
A diagonal band in the first quadrant is one big corner triangle minus a small one.
10.G-GPE.B.7Change Focus Count The ComplementSubtract the disk and round
Subtracting the disk and rounding gives 51, choice (C).
Punching the obtuse disk out of the acute strip leaves exactly the region asked for.
7.G.B.4Change Focus Count The ComplementEach angle being sharp is its own simple region — two half-planes and the outside of one circle — so the answer is just the area left after they overlap.
- Test each angle with a dot product
- Angle A cuts a half-plane
- Angle B cuts the other half-plane
- Angle C excludes a disk
- Recognise the circle on AB
- Check the disk fits inside the strip
- Area of the strip
- Subtract the disk and round
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