AMC 10 · 2018 · #22
Grade 11 geometry-2dPick an answer.
Tool #4 (Introduce a Variable): writing z = a + bi turns each complex equation into two real equations, which is the only way to get coordinates for the vertices. Tool #15 (Organize in More Ways): the two equations from matching parts lead to a quartic, but comparing lengths supplies a third equation for free, and all three are linear in a² and b² — the same information reorganized is much easier to solve. Tool #7 (Subproblems): solve the two square-root problems separately, then treat the area as its own geometry problem. Tool #1 (Draw a Diagram): plotting the four points shows they come in ± pairs, which fixes both the correct vertex order and the fastest area formula.
Split z into real and imaginary parts
Split into real and imaginary parts.
A complex number is really a pair of real numbers, so an equation between two of them is secretly two equations at once.
11.N-CN.A.1Introduce A VariableCompare lengths for a free third equation
Comparing sizes hands over a third equation for free.
Reading the same equation as a statement about lengths costs nothing and replaces a quartic with a two-by-two linear system.
11.N-CN.A.3Organize Information In More WaysSolve for the first pair of roots
Three equations fix the first pair of roots.
Add the two equations to erase b², subtract them to erase a² — the sum-and-difference pair is built for this.
9.A-REI.C.6Introduce A VariableRun the same routine on the second equation
The second equation follows the same routine.
The second equation is the same machine with smaller numbers, so reuse the routine instead of reinventing it.
11.N-CN.A.2Identify SubproblemsOpposite pairs force a parallelogram
Opposite pairs become the diagonals.
Square roots of a complex number always come in ± pairs, so the origin is automatically where the diagonals cross.
Square roots of a complex number always come in opposite pairs, so the origin is where the diagonals cross.
▸ Why?
A number and its opposite have the same square, so the two roots are mirror images through the origin.
▸ Why?
A complex number is a point in the plane, so opposite roots really are two points straddling the origin.
Four triangles of equal area
The four triangles have equal areas.
A parallelogram's diagonals split it into four triangles of equal area, so doing one triangle well finishes the whole shape.
10.G-GPE.B.7Identify SubproblemsSimplify into the required form
Tidying and adding gives 20.
The squarefree condition is a formatting rule whose whole job is to pin down one single way of writing the area.
11.N-RN.A.2Organize Information In More WaysTo square-root a complex number, set z = a + bi and match real parts, imaginary parts, and lengths — three equations that are linear in a² and b² — and since the four roots come in ± pairs they form a parallelogram whose diagonals cross at the origin, with area 2|x₁y₂ - x₂y₁| = 6√(2) - 2√(10).
- Split z into real and imaginary parts
- Compare lengths for a free third equation
- Solve for the first pair of roots
- Run the same routine on the second equation
- Opposite pairs force a parallelogram
- Four triangles of equal area
- Simplify into the required form