AMC 10 · 2009 · #10
Grade 8 geometry-2dA 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?
Pick an answer.
AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A $5$-meter flagpole snaps at a point $x$ meters above the ground. The upper part stays attached at the break and its tip reaches the ground $1$ meter from the base. Find the snap height $x$.
Givens: The flagpole is $5$ meters tall to start.; It snaps at a point $x$ meters above the ground, leaving a standing stump of height $x$.; The broken upper part stays hinged at the break, so its length is $5-x$ meters.; The tip of the broken part touches the ground $1$ meter from the base.; Answer choices: (A) $2.0$, (B) $2.1$, (C) $2.2$, (D) $2.3$, (E) $2.4$.
Unknowns: The height $x$ (in meters) where the flagpole snapped.
Understand
Restated: A $5$-meter flagpole snaps at a point $x$ meters above the ground. The upper part stays attached at the break and its tip reaches the ground $1$ meter from the base. Find the snap height $x$.
Givens: The flagpole is $5$ meters tall to start.; It snaps at a point $x$ meters above the ground, leaving a standing stump of height $x$.; The broken upper part stays hinged at the break, so its length is $5-x$ meters.; The tip of the broken part touches the ground $1$ meter from the base.; Answer choices: (A) $2.0$, (B) $2.1$, (C) $2.2$, (D) $2.3$, (E) $2.4$.
Plan
Primary tool: #1 Draw a Diagram
Secondary: #13 Convert to Algebra, #3 Eliminate Possibilities
The words hide a plain shape: a leaning broken pole is really a right triangle. Tool #1 (Draw a Diagram) surfaces that triangle and names its three sides in terms of $x$. Tool #13 (Convert to Algebra) then turns the right angle into the Pythagorean equation, and solving it pins down $x$. Tool #3 (Eliminate Possibilities) is a backup: because every choice is a clean decimal, the sides should form a scaled whole-number right triangle, which alone points to the answer.
Execute — Answer: E
6.EE.B.6 Step 1 Draw the leaning pole
- The standing stump rises straight up $x$ meters.
- The broken top hangs from the break down to the ground, landing $1$ meter from the base.
- The pole was $5$ meters and nothing is lost, so the broken top has length $5-x$.
- The vertical stump and the horizontal ground meet at a right angle, so the stump, the $1$-meter ground gap, and the broken top form a right triangle: the two legs are $x$ and $1$, and the slanted broken top is the hypotenuse $5-x$.
💡 A quick sketch turns a snapped pole into an ordinary right triangle you already know how to handle.
8.G.B.7 Step 2 Apply the Pythagorean theorem
- In a right triangle the squares of the two legs add up to the square of the hypotenuse.
- The legs are $x$ and $1$ and the hypotenuse is $5-x$, so the three side lengths must satisfy $x^2 + 1^2 = (5-x)^2$.
💡 The right angle is the key that unlocks $a^2+b^2=c^2$, tying the three side lengths into one equation.
8.EE.C.7 Step 3 Solve for x
- Expand the right side: $(5-x)^2 = 25 - 10x + x^2$.
- The equation becomes $x^2 + 1 = 25 - 10x + x^2$.
- The $x^2$ terms are on both sides, so they cancel, leaving $1 = 25 - 10x$.
- Then $10x = 24$, so $x = 2.4$.
- That is choice (E).
💡 Squaring the hypotenuse brings back an $x^2$ that matches the leg's $x^2$, so they cancel and the hard-looking equation collapses to a one-step linear one.
6.EE.B.6 The standing stump rises straight up $x$ meters. The broken top hangs from the b 8.G.B.7 In a right triangle the squares of the two legs add up to the square of the hypo 8.EE.C.7 Expand the right side: $(5-x)^2 = 25 - 10x + x^2$. The equation becomes $x^2 + 1 Review
Reasonableness: Test $x=2.4$: the legs are $2.4$ and $1$, and the broken top is $5-2.4=2.6$. Then $2.4^2 + 1^2 = 5.76 + 1 = 6.76 = 2.6^2$, so the triangle closes exactly. The break sits a little below the middle of the $5$-meter pole, which fits: the top piece ($2.6$ m) has to be long enough to lean over and touch the ground, so it should be a bit longer than the stump.
Alternative: Skip the algebra by spotting a Pythagorean triple. The sides $1$, $2.4$, $2.6$ are the whole-number triple $5$, $12$, $13$ shrunk by a factor of $5$. Since every answer choice is a clean decimal, the stump, gap, and broken top should form a scaled whole-number right triangle; the $5$-$12$-$13$ triangle whose short leg equals the $1$-meter gap forces the stump to be $2.4$ — choice (E).
CCSS standards used (min grade 8)
6.EE.B.6Use variables to represent numbers and write expressions for a real-world problem (Labeling the stump as $x$ and writing the broken top as $5-x$ from the total length $5$.)8.G.B.7Apply the Pythagorean theorem to determine unknown side lengths in right triangles (Setting up $x^2 + 1^2 = (5-x)^2$ from the right triangle formed by the stump, ground gap, and broken top.)8.EE.C.7Solve linear equations in one variable (Cancelling the $x^2$ terms and solving $1 = 25 - 10x$ to get $x = 2.4$.)
⭐ When something tall snaps and leans to the ground, draw the right triangle it makes — the leaning piece is the hypotenuse, and the Pythagorean theorem does the rest.
⭐ When something tall snaps and leans to the ground, draw the right triangle it makes — the leaning piece is the hypotenuse, and the Pythagorean theorem does the rest.
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