AMC 10 · 2017 · #17

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length
coordinate-geometrypythagorean-theoreminteger-pythagorean-triples extremal-constructionsystematic-enumeration ↑ Prerequisites: coordinate-geometry
📏 Medium solution 💡 3 insights
Problem
Four different points P, Q, R, S have integer coordinates and lie on the circle x² + y² = 25. The chord lengths PQ and RS are both irrational numbers. Find the largest possible value of the ratio PQ/RS.

Pick an answer.

(A)
3
(B)
5
(C)
$3\sqrt{5}$
(D)
7
(E)
$5\sqrt{2}$

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Extreme Principle

A ratio is largest when its top is as big as possible and its bottom as small as possible, so this is a boundary problem: maximize PQ and minimize RS separately. First list every lattice point on the circle (a finite, small set), then push each chord to its extreme while keeping both lengths irrational.

1STEP 1

List the lattice points

Solve x² + y² = 25 over the integers; allowing both signs on the pairs (0,5), (3,4), (4,3) gives twelve points.

(± 5, 0), (0, ± 5), (± 3, ± 4), (± 4, ± 3)
2STEP 2

When is a chord irrational?

A chord's length is the square root of a sum of squared coordinate gaps, irrational exactly when that sum is not a perfect square.

PQ = √((Δ x)² + (Δ y)²)
3STEP 3

Make RS as small as possible

Neighbors like (3, 4) and (4, 3) give the tightest gap, so the smallest irrational chord is RS = √2.

RS = √((4-3)² + (3-4)²) = √(2)
4STEP 4

Make PQ as large as possible

The diameter (10) is rational, so the longest irrational chord joins near-opposite points like (-3, 4) and (4, -3): PQ = 7√2.

PQ = √((4-(-3))² + (-3-4)²) = √(98) = 7√(2)
5STEP 5

Form the ratio

The matching √2 factors cancel, so PQ/RS = 7; the four points are distinct, making this ratio attainable — answer (D).

PQ/RS = 7√(2)/√(2) = 7
Answer
7
Both chosen chords are genuinely irrational: RS = sqrt(2) is about 1.41 and PQ = 7*sqrt(2) is about 9.90, which is under the diameter 10, so it is a real chord. The four points are distinct. The ratio 7 is one of the answer choices, and pushing the bottom to its floor and the top to its ceiling cannot be beaten.
💡Key takeaway

To make a ratio as big as possible, make the top as large as you can and the bottom as small as you can, then watch the matching square roots cancel.

  • List the lattice points
  • When is a chord irrational?
  • Make RS as small as possible
  • Make PQ as large as possible
  • Form the ratio

More like this

Same archetype — closest grade level first.