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2009 AMC 10B Problem 10

2009 AMC 10B Problem 10AMC 10 Step by Step

Name: Date: September 13, 2026

Triangles: Area & Pythagorean · difficulty 2 of 5 · about 2 min · position P1-10

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 $x$ ?

A)
$2.0$
B)
$2.1$
C)
$2.2$
D)
$2.3$
E)
$2.4$

Answer

Key insight

The broken piece is the hypotenuse 5 - x of a right triangle with legs x and 1, and the x squared terms cancel to leave a linear equation.

Solution

The stump is vertical with height $x$ ; the upper part has length $5 - x$ and leans from the top of the stump to a point on the ground $1$ meter from the base. These form a right triangle with legs $x$ and $1$ and hypotenuse $5 - x$ :
$x^2 + 1 = (5 - x)^2 = 25 - 10x + x^2.$
The $x^2$ terms cancel, leaving $1 = 25 - 10x$ , so $10x = 24$ and $x = 2.4$ .

Check: legs $2.4$ and $1$ , hypotenuse $2.6$ ; indeed $2.4^2 + 1 = 5.76 + 1 = 6.76 = 2.6^2$ .

The answer is $\boxed{\textbf{(E)}\ 2.4}$ .

Why this works

A "broken pole" is a right triangle whose hypotenuse plus one leg is a known total. Because both sides of the Pythagorean equation contain $x^2$ , the equation is really linear, so there is no need for the quadratic formula. Recognizing the $5$ - $12$ - $13$ shape ( $2.4 : 1 : 2.6 = 12 : 5 : 13$ ) confirms the result.

The trap

Taking the fallen piece to have length x, or setting the hypotenuse equal to 5 instead of 5 - x.

Common mistakes

Taking the fallen piece to have length x, or setting the hypotenuse equal to 5 instead of 5 - x.
Expanding $(5 - x)^2$ as $25 - x^2$ or $25 + x^2$ .

Techniques

Set up the equation/formula and compute; no special trick needed