AMC 10 · 2014 · #18

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
coordinate-geometrypythagorean-theorem convert-to-algebraspatial-visualization ↑ Prerequisites: coordinate-geometry
📏 Medium solution 💡 2 insights
Problem
A square sits in the coordinate plane, tilted so that its sides are not horizontal or vertical. Its four corners have y-coordinates 0, 1, 4, and 5. Find the area of the square.

Pick an answer.

(A)
16
(B)
17
(C)
25
(D)
26
(E)
27

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The x-coordinates are hidden, so track only how much each side rises. Tool #4 (Introduce a Variable) names the run and rise of one side; because the next side is that side rotated 90°, its run and rise are the same two numbers swapped. Tool #1 (Draw a Diagram) turns the four heights into a clear bottom corner, a top corner, and two side corners. Tool #7 (Identify Subproblems) splits the work into two clean pieces: first find the two rises, then feed them into the Pythagorean theorem for the area.

1STEP 1

Sort the heights and place the corners

Sorted, the heights read 0, 1, 4, 5: the lowest corner is at 0, its diagonal opposite at 5, and the two side corners at 1 and 4.

heights sorted: 0, 1, 4, 5
2STEP 2

Name the rise of each side

From the bottom corner one side rises p and the other rises q; a 90° turn swaps run and rise, so the far corner rises p+q.

corner heights above bottom: p, q, p+q
3STEP 3

Match the rises to the given heights

The three corners above sit at p, q, and p+q, matching 1, 4, and 5. The largest must be the sum, so p+q=5 and {p,q}={1,4}.

p+q=5, {p,q}={1,4}
4STEP 4

Use the right triangle to get the area

A side is the hypotenuse over run 1 and rise 4, so the area is s² = 1² + 4² = 17 — choice (B).

s² = 1² + 4² = 17 → (B)
Answer
17
The total height of the square is 5, so the side length is a bit less than 5 (a tilted square is taller than its side). A side just under 5 has area just under 25, and 17 fits that — its square root is about 4.12. The tilt is real because {p,q}={1,4} are different, so the square genuinely leans. Choices 16 and 25 are exactly the one-leg mistakes (4² and 5²), and 17 survives the full Pythagorean count.
💡Key takeaway

For a tilted square, follow the up-and-down rise of two sides that meet at a corner; those two rises are the legs of a right triangle, and the side squared (the area) is just leg squared plus leg squared.

  • Sort the heights and place the corners
  • Name the rise of each side
  • Match the rises to the given heights
  • Use the right triangle to get the area

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