Difference between revisions of "2009 AMC 10B Problems/Problem 10"
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(Note that the resulting triangle is the well-known <math>5-12-13</math> right triangle, scaled by <math>1/5</math>.) | (Note that the resulting triangle is the well-known <math>5-12-13</math> right triangle, scaled by <math>1/5</math>.) | ||
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== Solution 2 == | == Solution 2 == | ||
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== Solution 3 == | == Solution 3 == | ||
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Let <math>AB</math> represent the flagpole in the diagram above. After the flagpole breaks at point <math>D</math>, its tip lies at point <math>C</math>. Since none of the flagpole is destroyed, we know that <math>DA=DC</math>. Therefore, triangle <math>\triangle ADC</math> is isosceles. | Let <math>AB</math> represent the flagpole in the diagram above. After the flagpole breaks at point <math>D</math>, its tip lies at point <math>C</math>. Since none of the flagpole is destroyed, we know that <math>DA=DC</math>. Therefore, triangle <math>\triangle ADC</math> is isosceles. | ||
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Since <math>AB = 5</math> and <math>BC = 1</math>, we have that <math>AD = \frac{5^2 + 1^2}{2 \cdot 5} = 2.6</math>, and thus <math>x = AB - AD = \boxed{2.4}</math>. | Since <math>AB = 5</math> and <math>BC = 1</math>, we have that <math>AD = \frac{5^2 + 1^2}{2 \cdot 5} = 2.6</math>, and thus <math>x = AB - AD = \boxed{2.4}</math>. | ||
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+ | ==Video Solution== | ||
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+ | There is currently no video solution on Youtube. A wrong link was posted here. | ||
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+ | (I sent a request to BeautyofMath so there may be one on his channel soon) | ||
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+ | ~ ssr-07 | ||
== See Also == | == See Also == |
Latest revision as of 18:12, 12 July 2023
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 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 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.
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
There is currently no video solution on Youtube. A wrong link was posted here.
(I sent a request to BeautyofMath so there may be one on his channel soon)
~ ssr-07
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.