AMC 10 · 2016 · #10

Grade 8 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Lattice or Non-Aligned Polygon Area

coordinate-geometrypythagorean-theoremdifference-of-squaresfactors coordinate-geometrysystematic-enumeration ↑ Prerequisites: coordinate-geometrypythagorean-theorem

📏 Medium solution 💡 3 insights

Problem

Four points built from two whole numbers bound a region of known area. Find their sum.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Draw a Diagram

The coordinates are given in terms of letters, so a sketch with sample numbers is the fastest way to see what kind of quadrilateral this is. Once the shape is pinned down, the area splits into small subproblems: measure the sides with the distance formula, check the corner is square, then multiply. That turns the area condition into one clean equation in a and b, and because a and b are integers the equation can be finished by listing factor pairs and crossing out the impossible ones.

1STEP 1

Plot the points and spot the symmetry

The figure is centred at the origin.

R = -P, S = -Q ⟹ midpoint of PR = midpoint of QS = (0,0)

2STEP 2

Measure the four sides

Opposite sides come out equal.

PQ = RS = (a-b)√(2), QR = SP = (a+b)√(2)

3STEP 3

Show every corner is a right angle

Every corner turns out to be a right angle.

PQ² + QR² = 2(a-b)² + 2(a+b)² = 4a² + 4b² = PR²

4STEP 4

Turn the area into one equation

The area becomes a plain product.

[(a-b)√(2)][(a+b)√(2)] = 2(a-b)(a+b) = 16 ⟹ (a-b)(a+b) = 8

5STEP 5

List the factor pairs of 8

Only two factor pairs are possible.

8 = 1 × 8 = 2 × 4, a+b ＞ a-b ＞ 0

6STEP 6

Eliminate one pair and solve

Parity leaves one, giving 4, choice (B).

a+b = 4 ; a-b = 2 ⟹ a = 3, b = 1, a+b = 4

Answer

Put the numbers back in. With a=3, b=1 the vertices are P(3,1), Q(1,3), R(-3,-1), S(-1,-3). The sides measure 2√(2) and 4√(2), and 2√(2) · 4√(2) = 16, matching the stated area, and both diagonals measure 2√(10), as a rectangle requires. A search over all integer pairs with a ＞ b ＞ 0 produces this one solution and no other, so a+b=4 is forced rather than merely possible. The bigger choices (D) 12 and (E) 13 would need a²-b²=8 with much larger a, which is impossible since a²-b² ≥ (a+b) grows past 8 immediately.

💡Key takeaway

When a figure's corners come in opposite pairs like (a,b) and (-a,-b), it is centered on the origin, and measuring its sides usually turns the area into a small factoring puzzle.

Plot the points and spot the symmetry
Measure the four sides
Show every corner is a right angle
Turn the area into one equation
List the factor pairs of 8
Eliminate one pair and solve