2009 AMC 10B Problems/Problem 10
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 ?
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 .)
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.
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 .
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