AMC 10 · 2004 · #8
Grade 8 geometry-2dMinneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
Pick an answer.
AMC 10 2004 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: An airport sits $8$ miles southwest of downtown St. Paul and $10$ miles southeast of downtown Minneapolis. Using these two compass distances, find which whole number of miles is closest to the straight-line distance between the two downtowns.
Givens: The airport is $8$ miles southwest of St. Paul, so St. Paul is $8$ miles northeast of the airport.; The airport is $10$ miles southeast of Minneapolis, so Minneapolis is $10$ miles northwest of the airport.; Answer choices: (A) $13$, (B) $14$, (C) $15$, (D) $16$, (E) $17$
Unknowns: The distance in miles between downtown St. Paul and downtown Minneapolis.
Understand
Restated: An airport sits $8$ miles southwest of downtown St. Paul and $10$ miles southeast of downtown Minneapolis. Using these two compass distances, find which whole number of miles is closest to the straight-line distance between the two downtowns.
Givens: The airport is $8$ miles southwest of St. Paul, so St. Paul is $8$ miles northeast of the airport.; The airport is $10$ miles southeast of Minneapolis, so Minneapolis is $10$ miles northwest of the airport.; Answer choices: (A) $13$, (B) $14$, (C) $15$, (D) $16$, (E) $17$
Plan
Primary tool: #1 Draw a Diagram
Secondary: #17 Visualize Spatial Relationships, #3 Eliminate Possibilities
The problem is really about directions, so the first move is to place the airport on a diagram and draw the two segments to the downtowns (Tool #1). Reading the compass words carefully (Tool #17) shows the segment to St. Paul points northeast and the segment to Minneapolis points northwest — and those two directions are perpendicular. That turns the picture into a right triangle whose legs are $8$ and $10$ and whose hypotenuse is the distance we want, so the Pythagorean theorem finishes it. Because the true distance is not a whole number, Tool #3 (Eliminate Possibilities) is used at the end to pick the closest choice.
Execute — Answer: A
4.G.A.1 Step 1 Draw the two roads from the airport
- Put the airport at the center of a compass.
- St.
- Paul is $8$ miles to the northeast of it, and Minneapolis is $10$ miles to the northwest of it.
- Northeast points halfway between north and east; northwest points halfway between north and west.
- These two rays sit on opposite sides of due north, each tilted $45^\circ$ away from it, so the angle between them at the airport is $45^\circ + 45^\circ = 90^\circ$.
- The airport is therefore the square corner of a right triangle whose two legs are the $8$-mile and $10$-mile segments.
💡 Northeast and northwest lean away from north by the same amount on opposite sides, so together they open up a right angle.
8.G.B.7 Step 2 Apply the Pythagorean theorem
- The distance between the two downtowns is the hypotenuse of that right triangle, opposite the right angle at the airport.
- With legs $8$ and $10$, the Pythagorean theorem gives the hypotenuse $d$ from $d^2 = 8^2 + 10^2 = 64 + 100 = 164$, so $d = \sqrt{164}$ miles.
💡 In a right triangle the hypotenuse squared equals the sum of the two leg squares, so squaring the legs and adding gives the missing side.
8.NS.A.2 Step 3 Estimate the square root and pick the choice
- Now find which whole number $\sqrt{164}$ is nearest.
- Since $12^2 = 144$ and $13^2 = 169$, the value $\sqrt{164}$ lies between $12$ and $13$.
- Because $164$ is much closer to $169$ than to $144$, the root is close to $13$; checking $12.8^2 = 163.84$ confirms $\sqrt{164} \approx 12.8$.
- The nearest choice is $13$, so the answer is (A).
💡 Squaring the whole numbers around the root tells you which one it sits closest to.
4.G.A.1 Put the airport at the center of a compass. St. Paul is $8$ miles to the northea 8.G.B.7 The distance between the two downtowns is the hypotenuse of that right triangle, 8.NS.A.2 Now find which whole number $\sqrt{164}$ is nearest. Since $12^2 = 144$ and $13^ Review
Reasonableness: The hypotenuse of a right triangle must be longer than either leg but shorter than their sum, so the distance sits between $10$ and $8+10=18$ miles. The estimate $\sqrt{164}\approx 12.8$ falls in that window and is only a little more than the longer leg of $10$, which is exactly what a right triangle with legs $8$ and $10$ should give. Rounding to the nearest listed value gives $13$, choice (A).
Alternative: Scale the triangle to a familiar one: legs $8$ and $10$ share the ratio $4:5$ with the $3$-$4$-$5$ family only loosely, but note the triangle is exactly twice a right triangle with legs $4$ and $5$, whose hypotenuse is $\sqrt{41}\approx 6.4$. Doubling gives $2\times 6.4 = 12.8$ miles, matching $\sqrt{164}$ and again pointing to (A).
CCSS standards used (min grade 8)
4.G.A.1Draw and identify points, lines, rays, angles, and perpendicular lines (Reading the compass directions to see that the northeast and northwest segments meet at a right angle at the airport.)8.G.B.7Apply the Pythagorean Theorem to find unknown side lengths in right triangles in real-world problems (Computing the hypotenuse $\sqrt{8^2+10^2}=\sqrt{164}$ as the distance between the two downtowns.)8.NS.A.2Use rational approximations of irrational numbers to compare and locate them (Estimating $\sqrt{164}\approx 12.8$ and choosing the nearest answer, $13$.)
⭐ When two compass directions meet at a right angle, the straight-line distance across is the hypotenuse — square the two legs, add, and take the square root.
⭐ When two compass directions meet at a right angle, the straight-line distance across is the hypotenuse — square the two legs, add, and take the square root.
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