AMC 10 · 2006 · #20

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length
coordinate-geometrypythagorean-theoremslope-intercept convert-to-algebra ↑ Prerequisites: coordinate-geometry
📏 Medium solution 💡 3 insights
Problem
In rectangle ABCD the corners are named in order, so sides AB and AD meet at a right angle at corner A. Three corners are known on the coordinate plane: A=(6,-22), B=(2006,178), and D=(8,y) for some integer y. What is the area of rectangle ABCD?

Pick an answer.

(A)
4000
(B)
4040
(C)
4400
(D)
40,000
(E)
40,400

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The corners are given as coordinates, so Tool #1 (Draw a Diagram) turns the problem into a picture: two sides, AB and AD, shoot out from corner A and must meet at a right angle. Reading each side as a horizontal-and-vertical step makes the right angle usable. Tool #4 (Introduce a Variable) handles the unknown height y: the perpendicular condition gives one equation that pins it down. Tool #7 (Identify Subproblems) then splits the area into two clean pieces — measure the length of AB, measure the length of AD — because area is just their product. A bonus of reading the sides as steps: the perpendicular side lengths multiply to a whole number even though each side is irrational.

1STEP 1

Read each side as a horizontal-vertical step

From A=(6,-22), the step to B is right 2000, up 200; the step to D is right 2, up y+22.

AB: (2000, 200) AD: (2, y+22)
2STEP 2

Turn the corner 90 degrees to find y

The square corner swaps the two numbers and flips a sign: AB's (2000,200) turns to direction (1,-10), so AD=(2,-20) and y=-42.

(2, y+22)=2·(1,-10)=(2,-20) → y+22=-20 → y=-42
3STEP 3

Measure both sides with the Pythagorean theorem

Each side is a hypotenuse: AB=√(2000²+200²)=100√404 and AD=√(2²+20²)=√404.

AB=√(2000²+200²)=100√(404), AD=√(2²+20²)=√(404)
4STEP 4

Multiply the two sides for the area

The shared roots cancel: area = AB·AD = 100√404·√404 = 100·404 = 40,400, choice (E).

Area=AB · AD=100√(404)·√(404)=100·404=40,400 → (E)
Answer
40,400
The horizontal gap alone from A to B is 2000 and the vertical gap is 200, so AB is a bit over 2000; side AD is short, a bit over 20. Their product should be a little above 2000×20=40,000, and 40,400 sits exactly there — the two nearby choices (D) 40,000 and (E) 40,400 are the only ones in that range, and the tiny extra from the diagonal slant makes (E) the right one. It also checks that y=-42 is an integer, as the problem promised.
💡Key takeaway

Read each rectangle side as a right-and-up step; a square corner turns one step's numbers a quarter turn (swap and flip a sign), and multiplying the two side lengths lets the matching square roots cancel into a clean 40,400.

  • Read each side as a horizontal-vertical step
  • Turn the corner 90 degrees to find y
  • Measure both sides with the Pythagorean theorem
  • Multiply the two sides for the area

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