2009 AMC 10B Problems/Problem 10
Problem
Captain Underpants got a wedgie. Calculate how much he will fall if the pole gets knocked over. Will he die. Are you an idiot?
Solution 1
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 Captain Underpants, 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.
Solution 3
Let represent the flagpole in the diagram above. After the flagpole breaks at point
, its tip lies at point
. Since none of the flagpole is destroyed, we know that
. Therefore, triangle
is isosceles.
Draw the altitude . Since
is isosceles, we know that
. Also note that
. Therefore,
Since and
, we have that
, and thus
.
Video Solution
https://www.youtube.com/watch?v=dQw4w9WgXcQ
~Anonymous23
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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