Difference between revisions of "2009 AMC 10B Problems/Problem 10"
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== Solution 2 == | == Solution 2 == | ||
− | A right triangle is formed with the bottom of the flagpole, the snapped part, and the ground. One leg is of length <math>1</math> and the other is length <math>x</math>. By the [[ | + | A right triangle is formed with the bottom of the flagpole, the snapped part, and the ground. One leg is of length <math>1</math> and the other is length <math>x</math>. By the [[]], we know that <math>\sqrt{x^2+1^2}</math> must be the length of the snapped part of the flagpole. Observe that all the answer choices are rational. If <math>x</math> is rational, <math>5-x</math>, which is the snapped part, must also be rational. Therefore, <math>1, x, 5-x</math> must form a scaled Pythagorean triple. We know that <math>10, 24, 26</math> is a Pythagorean triple, so the corresponding answer must be <math>1, 2.4, 2.6</math>. Adding together the <math>x</math> and the snapped part, this does indeed equal <math>5</math>, so our solution is done. |
== Solution 3 == | == Solution 3 == |
Revision as of 20:24, 6 June 2021
Problem
captain Under pants 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 [[]], 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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.