AMC 10 · 2016 · #17
Grade 10 geometry-2d
Pick an answer.
Chasing where two bisector lines of the big triangle cross a third line is messy. But the altitude already cuts △ ABC into two right triangles, △ ABH and △ ACH, and AH is a full side of each. Inside △ ABH the bisector from B is a cevian to side AH; inside △ ACH the bisector from C is a cevian to the same side AH. That is exactly the setup the angle bisector theorem measures, so the one hard problem becomes two easy independent ones (Tool #7). Getting there needs the three side lengths of each right triangle, which comes from naming BH and AH and using the Pythagorean theorem twice (Tool #4). Finally the two ratios have to be re-expressed as distances from the same endpoint A before they can be subtracted (Tool #15).
Name the pieces the altitude makes
The altitude splits the base into whole pieces.
One altitude builds two right triangles that share the same leg, so the two Pythagorean statements are locked together.
8.G.B.7Introduce A VariableSubtract to locate the foot
Subtracting locates the foot at 2.
The shared leg is the unknown you do not want, and subtracting the two equations erases it in one move.
8.EE.C.8Introduce A VariableRead BD inside △ ABH
One bisector works inside a smaller triangle.
A bisector does not care which triangle you choose to look at — inside the small right triangle it still bisects the same angle.
10.G-SRT.B.5Identify SubproblemsRead CE inside △ ACH
The other does the same on its side.
The same move works on the other half of the altitude, using that half's own two side lengths.
10.G-SRT.B.5Identify SubproblemsTurn each ratio into a distance from A
Each ratio becomes a distance down the altitude.
A ratio says how many equal shares each part gets, so each part is that fraction of the whole segment.
A ratio says how many equal shares each part gets, so each part is that fraction of the whole segment.
▸ Why?
Each share is one equal piece of the whole, so counting pieces names the length directly.
▸ Why?
The bisector splits the far side in the ratio of the two sides beside it, which is what fixes the shares.
Subtract the two distances
Subtracting gives 8√5/15, choice (D).
Two marks measured from the same end of a segment differ by exactly the gap between them.
7.NS.A.1Organize Information In More WaysDrop the altitude first: each angle bisector then lives inside a small right triangle, where it cuts the altitude in the ratio of that triangle's own two known sides.
- Name the pieces the altitude makes
- Subtract to locate the foot
- Read BD inside △ ABH
- Read CE inside △ ACH
- Turn each ratio into a distance from A
- Subtract the two distances
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Same archetype — closest grade level first.
