2009 AMC 10B Problems/Problem 10

Problem

A flagpole is originally $5$ meters tall. A hurricane snaps the flagpole at a point $x$ meters above the ground so that the upper part, still attached to the stump, touches the ground $1$ meter away from the base. What is $x$ ?

$\text{(A) } 2.0 \qquad \text{(B) } 2.1 \qquad \text{(C) } 2.2 \qquad \text{(D) } 2.3 \qquad \text{(E) } 2.4$

Solution

The broken flagpole forms a right triangle with legs $1$ and $x$ , and hypotenuse $5-x$ . The Pythagorean theorem now states that $1^2 + x^2 = (5-x)^2$ , hence $10x = 24$ , and $x=\boxed{2.4}$ .

(Note that the resulting triangle is the well-known $5-12-13$ right triangle, scaled by $1/5$ .)

Solution #2

A right triangle is formed with the bottom of the flagpole, the snapped part, and the ground. One leg is of length $1$ and the other is length $x$ . By the Pythagorean theorem, we know that $\sqrt{x^2+1^2}$ must be the length of the snapped part of the flagpole. Observe that all the answer choices are rational. If $x$ is rational, $5-x$ , which is the snapped part, must also be rational. Therefore, $1, x, 5-x$ must form a scaled Pythagorean triple. We know that $10, 24, 26$ is a Pythagorean triple, so the corresponding answer must be $1, 2.4, 2.6$ . Adding together the $x$ and the snapped part, this does indeed equal $5$ , so our solution is done.

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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2009 AMC 10B Problems/Problem 10

Contents

Problem

Solution

Solution #2

See Also