AMC 10 · 2009 · #10
Grade 8 geometry-2dPick an answer.
AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The words hide a plain shape: a leaning broken pole is really a right triangle. Tool #1 (Draw a Diagram) surfaces that triangle and names its three sides in terms of x. Tool #13 (Convert to Algebra) then turns the right angle into the Pythagorean equation, and solving it pins down x. Tool #3 (Eliminate Possibilities) is a backup: because every choice is a clean decimal, the sides should form a scaled whole-number right triangle, which alone points to the answer.
Draw the leaning pole
The stump is x, the ground gap is 1, and nothing is lost, so the leaning top is 5-x — a right triangle with that as hypotenuse.
A quick sketch turns a snapped pole into an ordinary right triangle you already know how to handle.
6.EE.B.6Draw A DiagramApply the Pythagorean theorem
Legs x and 1 with hypotenuse 5-x, so the Pythagorean theorem gives x² + 1² = (5-x)².
The right angle is the key that unlocks a²+b²=c², tying the three side lengths into one equation.
The right angle is the key that ties the three side lengths into one equation.
▸ Why?
With a right angle the two legs squared add to the longest side squared.
▸ Why?
Squaring and simplifying both sides keeps the statement true, so the equation can be solved safely.
Solve for x
Expanding gives x² + 1 = 25 - 10x + x²; the x² cancels, leaving 10x = 24, so x = 2.4 — choice (E).
Squaring the hypotenuse brings back an x² that matches the leg's x², so they cancel and the hard-looking equation collapses to a one-step linear one.
8.EE.C.7Convert To AlgebraWhen something tall snaps and leans to the ground, draw the right triangle it makes — the leaning piece is the hypotenuse, and the Pythagorean theorem does the rest.
- Draw the leaning pole
- Apply the Pythagorean theorem
- Solve for x