AMC 10 · 2012 · #10
Grade 9 geometry-2dPick an answer.
"Points of intersection" is geometry's phrase for "common solutions of a system", so Tool #13 (Convert to Algebra) turns the whole question into two equations in two unknowns. Tool #15 (Organize Information in More Ways) supplies the key move: both equations contain y only as y², and the second one carries 9y² while the first carries y² — so scaling the first equation by 9 makes the two y-terms identical and a single subtraction erases y completely, leaving one quadratic in x. That quadratic proves only necessity: it says which x-values a shared point could have. Tool #3 (Eliminate Possibilities) closes the gap by recovering y for each surviving x and substituting every resulting point back into both original equations — that substitution is what upgrades "can only be among these" into "is exactly these". Tool #1 (Draw a Diagram) then plots the survivors so the base and the height can be read straight off the coordinates, and Tool #7 (Identify Subproblems) keeps the two halves apart: first pin down the points, then compute the area. Worth noticing what this plan never uses: no formula for a circle, no formula for an ellipse, no conic theory at all — just a system of two equations.
A vertex solves both equations
A corner must satisfy both equations.
"On both curves" is just another way of saying "makes both equations true at the same time".
9.A-CED.A.3Convert To AlgebraScale, then subtract to erase y
Scaling and subtracting erases one variable.
Lining up the y² coefficients lets one subtraction wipe y out of the problem.
Lining up the coefficients on one variable lets a single subtraction wipe it out of the problem.
▸ Why?
Scaling a true equation and subtracting another keeps the result true, so the move is legitimate.
▸ Why?
Both equations then carry the identical term, so the subtraction removes it entirely.
Factor the quadratic in x
What remains factors into two candidates.
With y gone, what is left is an ordinary quadratic, and its two roots are the only x-values a shared point can have.
9.A-REI.B.4Convert To AlgebraRecover y and check every candidate
One candidate gives a single point, the other gives two.
Narrowing to candidates only proves "no others exist"; putting them back into both equations is what proves these ones do.
8.EE.A.2Eliminate PossibilitiesThree points, one triangle
Three points make a triangle.
Choosing the vertical side as the base makes both the base and the height plain subtractions of coordinates.
6.G.A.1Draw A DiagramArea of the triangle
Its area is 27, choice (B).
Once the base is vertical and the third corner is straight out to the side, the area is a single multiplication.
6.G.A.1Identify SubproblemsIntersection points are just the solutions of both equations at once: match the y² terms so one subtraction erases y, solve the quadratic that is left, then plug each candidate back into both equations to prove it is real — the three survivors make a triangle with base 6 and height 9.
- A vertex solves both equations
- Scale, then subtract to erase y
- Factor the quadratic in x
- Recover y and check every candidate
- Three points, one triangle
- Area of the triangle