AMC 10 · 2006 · #15
Grade 8 geometry-2d
Pick an answer.
A concave hexagon looks scary, but Tool #7 (Identify Subproblems) turns it into pieces we already know. The line OP splits the figure into two mirror-image halves, and each half is just the quadrilateral OADP — a right trapezoid, because both radii OA and PD stand perpendicular to the same tangent line AD and are therefore parallel. So the plan is: (1) draw the radii to the tangent points and mark the right angles (Tool #1, Draw a Diagram); (2) build a right triangle to find the tangent length AD with the Pythagorean theorem; (3) find the area of the right trapezoid OADP; (4) double it, since symmetry (Tool #17) makes the lower half congruent to the upper half. Two known shapes replace one strange one.
Draw the radii and spot the parallel sides
Radii to the same tangent are parallel, so a trapezoid appears.
Both radii lean against the same tangent line at right angles, so they point the same way and are parallel.
Both radii lean against the same tangent line at right angles, so they point the same way and never meet.
▸ Why?
A radius drawn to a tangent point meets the tangent square on, fixing its direction exactly.
▸ Why?
Two lines with the same direction keep a constant gap and never cross, which is what makes the shape a trapezoid.
Build a right triangle to find AD
A right triangle inside gives the tangent length 4√(2).
Chopping off a rectangle leaves one clean right triangle whose missing leg is exactly the tangent length.
8.G.B.7Identify SubproblemsFind the area of trapezoid OADP
The trapezoid's area is 12√(2).
A trapezoid's area is just the average width times how far apart the parallel edges sit.
6.G.A.1Identify SubproblemsDouble it using the symmetry
Symmetry doubles it to 24√(2), choice (B).
Mirror symmetry across OP means the bottom half is a free copy of the top half, so just double.
4.G.A.3Visualize Spatial RelationshipsA radius always meets its tangent at a right angle, so a tangent problem secretly hides right triangles and simple trapezoids — find them, and the Pythagorean theorem hands you the missing length.
- Draw the radii and spot the parallel sides
- Build a right triangle to find AD
- Find the area of trapezoid OADP
- Double it using the symmetry
More like this
Same archetype — closest grade level first.
