AMC 10 · 2012 · #12

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length
pythagorean-theoremisosceles-triangleestimation identify-subproblems ↑ Prerequisites: pythagorean-theorem
📏 Medium solution 💡 2 insights
Problem
Point B is due east of point A, and point C is due north of point B. The distance AC is 10√(2), and ∠ BAC = 45°. Point D is 20 meters due north of point C. The distance AD is between which two integers?

Pick an answer.

(A)
$30\ \text{and}\ 31$
(B)
$31\ \text{and}\ 32$
(C)
$32\ \text{and}\ 33$
(D)
$33\ \text{and}\ 34$
(E)
$34\ \text{and}\ 35$

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The problem is pure compass directions, so Tool #1 (Draw a Diagram) turns "east" and "north" into a picture where those two directions meet at a right angle at B — that single right angle is the key that unlocks everything. Tool #7 (Identify Subproblems) splits the work into two right triangles that share the leg AB: the small triangle ABC pins down the eastward distance AB, and the big triangle ABD gives AD. Finally Tool #3 (Eliminate Possibilities) squares the boundary integers to trap AD between the correct consecutive pair.

1STEP 1

Draw the directions as right angles

East is horizontal and north is vertical, so A → B → C turns a square corner at B, and B, C, D all sit on that one vertical line.

∠ ABC = 90°, B, C, D on one north line
2STEP 2

Find the eastward leg AB

Triangle ABC is right at B with a 45° angle, so its legs match: 2 AB² = (10√(2))² = 200, giving AB = BC = 10.

2 AB² = (10√(2))² = 200 → AB = BC = 10
3STEP 3

Build the big right triangle and get AD

Triangle ABD is right at B too, with legs AB = 10 and BD = BC + CD = 30, so AD² = 100 + 900 = 1000 and AD = √(1000).

AD² = 10² + 30² = 1000 → AD = √(1000)
4STEP 4

Trap the square root between integers

Square the neighbours: 31² = 961 ＜ 1000 ＜ 1024 = 32², so √(1000) sits between 31 and 32 — choice (B).

31² = 961 ＜ 1000 ＜ 1024 = 32² → 31 ＜ AD ＜ 32 (B)
Answer
31 and 32
A quick sanity pass: AD = √(1000) = 10√(10) ≈ 10 × 3.162 = 31.6, which indeed sits between 31 and 32. It also should be a touch more than the vertical leg BD = 30 (the hypotenuse always beats either leg), and 31.6 is just over 30 — consistent. The small eastward leg AB = 10 only nudges the length up from 30 by about 1.6, so pairs like (D) 33–34 or (E) 34–35 would be far too big.
💡Key takeaway

East and north make a right angle, so the 45° triangle gives AB = 10; then one big right triangle gives AD = √(10²+30²) = √(1000) ≈ 31.6, which lands between 31 and 32.

  • Draw the directions as right angles
  • Find the eastward leg AB
  • Build the big right triangle and get AD
  • Trap the square root between integers

More like this

Same archetype — closest grade level first.