AMC 10 · 2017 · #18
Grade 8 geometry-2dPick an answer.
Tool #1 (Draw a Diagram): the problem stacks a circle, a stretched diameter, and a perpendicular segment, so a labeled picture is the only way to see how the pieces line up. Tool #7 (Identify Subproblems): I split the work into 'find the big right triangle ADE', 'notice the small right triangle ACB hiding inside it', and 'turn the matching shapes into lengths'. Tool #4 (Introduce a Variable): I name the two legs AC and CB of the target triangle and let the similar triangles pin their lengths.
Set up the figure
Adding the pieces gives the whole base.
A clean picture with the right angle marked shows which lengths are legs of which right triangle.
4.G.A.1Draw A DiagramFind AE
The right triangle gives the hypotenuse.
The two known legs of a right triangle fix the hypotenuse through a²+b²=c².
8.G.B.7Identify SubproblemsSpot the similar triangles
The angle on the diameter is also right, so the triangles are similar.
An angle drawn on a diameter is always a right angle, which hands you a second matching right triangle.
An angle drawn on a diameter is always a right angle, which hands over a second matching right triangle.
▸ Why?
Every point of the circle is one radius from the centre, so the midpoint of the diameter is equally far from all three corners.
▸ Why?
With two triangles sharing an angle and each holding a right angle, all their matching sides sit in one ratio.
Scale the legs
The similarity ratio shrinks both legs.
Matching sides of similar triangles all shrink by the same factor, here the ratio of the two hypotenuses.
8.G.A.4Introduce A VariableCompute the area
Half the product of the legs is one hundred forty thirty-sevenths.
For a right triangle the two legs are the base and height, so the area is just half their product.
6.G.A.1Identify SubproblemsAny triangle whose longest side is a diameter has a right angle, so the little triangle ABC is just a scaled copy of the big right triangle ADE — match the sides, then take half of leg times leg to get area 140/37, choice (D).
- Set up the figure
- Find AE
- Spot the similar triangles
- Scale the legs
- Compute the area
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