2009 AMC 10B Problems/Problem 10
Contents
Problem
A flagpole is originally meters tall. A hurricane snaps the flagpole at a point
meters above the ground so that the upper part, still attached to the stump, touches the ground
meter away from the base. What is
?
Solution
The broken flagpole forms a right triangle with legs and
, and hypotenuse
. The Pythagorean theorem now states that
, hence
, and
.
(Note that the resulting triangle is the well-known right triangle, scaled by
.)
Solution #2
A right triangle is formed with the bottom of the flagpole, the snapped part, and the ground. One leg is of length and the other is length
. By the Pythagorean theorem, we know that
must be the length of the snapped part of the flagpole. Observe that all the answer choices are rational. If
is rational,
, which is the snapped part, must also be rational. Therefore,
must form a scaled Pythagorean triple. We know that
is a Pythagorean triple, so the corresponding answer must be
. Adding together the
and the snapped part, this does indeed equal
, so our solution is done.
See Also
2009 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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All AMC 10 Problems and Solutions |
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