Difference between revisions of "2009 AMC 10B Problems/Problem 10"
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Revision as of 16:27, 19 May 2021
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 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
size(300); pair A, B, C, D, E, x; A =(0, 5); B = (0, 0); C = (3, 0); D = (0, 1.6); E = (A+C)/2; x = (B+D)/2; draw(MP("A",A, (0, 1))--MP(P("C",C,(1, -1))--cycle); draw(C--MP("D",D, (-1, 0))--MP("E",E,(1, 1))); MP("x",x, (-1, 0)); draw(rightanglemark(A, B, C)); draw(rightanglemark(A, E, D)); draw(anglemark(B, A, C)); (Error making remote request. Unknown error_msg)
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
.
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.