AMC 10 · 2025 · #22
Grade 11 algebrageometry-2dPick an answer.
The question says "greatest possible," so Tool #14 (Extreme Principle) owns the finish: once the area is written as a single increasing function of |z|, the winner is whichever allowed z sits farthest from the origin, and on a circle that point is easy to name. Getting to that clean form is Tool #7 (Identify Subproblems): the common factor z splits the problem into "what shape is the triangle" and "how big is |z|," two questions that do not interfere. Tool #9 (Solve an Easier Related Problem) does the shape half by throwing z away entirely and measuring the fixed triangle 2, 1+i, 1-i, which is just three plotted points. Tool #1 (Draw a Diagram) makes that measurement a one-line base-times-height. Tool #15 (Organize Information in More Ways) handles the constraint: |4z-2|=1 says nothing useful until it is rearranged into center-and-radius form.
Pull the common factor z out
Pull out the common factor z.
The same z multiplies all three vertices, so it cannot change the shape — it can only stretch every length by the same factor |z|.
11.N-CN.A.3Identify SubproblemsAreas scale by the square of |z|
The area scales by the modulus squared.
Area is a length times a length, so scaling every length by |z| scales area by |z| twice over.
Area is a length times a length, so scaling every length by a factor scales the area by that factor twice over.
▸ Why?
Area lives in two directions at once, so it picks up the scale factor once for each of them.
▸ Why?
One common factor stretches every length together, so the shape is untouched and only the size changes.
Measure the fixed triangle
The fixed triangle has area 1.
Two of the three points share an x-coordinate, so the base is handed to you vertically and the height is just a horizontal gap.
10.G-GPE.B.7Draw A DiagramRead the constraint as a circle
The condition is a circle of radius one quarter about one half.
A modulus of a difference is a distance, so an equation of the form |z-c|=r is a circle in disguise.
10.G-GPE.A.1Organize Information In More WaysPush z to the far side of the circle
At the far point the modulus is three quarters, giving nine sixteenths.
To get as far as possible from the origin, first go to the center, then keep walking in the same direction for one more radius.
10.G-GPE.B.4Extreme PrincipleWhen one complex number multiplies every vertex, the shape stays put and only the size changes — so the area is a fixed number times |z|², and you just need the point of the circle farthest from the origin.
- Pull the common factor z out
- Areas scale by the square of |z|
- Measure the fixed triangle
- Read the constraint as a circle
- Push z to the far side of the circle