Coordinate Geometry (AIME / USAJMO / AMO)

Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.

Proof track: treat each problem as a full olympiad writeup on paper. Here you only enter the final answer (integer or short expression).

Coordinate Geometry: core idea

Optimal placement.

**(Proof track.)** Give a complete justification. Prove the triangle $(1,1),(7,3),(3,9)$ is right, then find its area.

Hint: Use coordinates to turn every claimed geometric fact into algebra. Answer: 24.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. Find the equation of the circle through $(0,0),(6,0),(0,8)$ and give its radius.
2. **(Proof track.)** Give a complete justification. The medians of triangle $(0,0),(6,0),(2,8)$ meet at $(a,b)$ . Find $a+b$ .
3. **(Proof track.)** Give a complete justification. A line through $(2,5)$ perpendicular to $2x-3y=7$ meets the $x$ -axis at $(a,0)$ . Find $a$ .
4. **(Proof track.)** Give a complete justification. Find the area of the triangle formed by $x=0,y=0,$ and $2x+3y=12$ .
5. **(Proof track.)** Give a complete justification. Show $(2,1),(8,5),(6,-1)$ is isosceles and find its perimeter.
6. **(Proof track.)** Give a complete justification. Find the point on $y=x^2$ closest to $(0,3)$ ; enter its $x$ -coordinate.
7. **(Proof track.)** Give a complete justification. The diagonals of a parallelogram have endpoints $(0,0),(8,6)$ and $(2,7),(6,-1)$ . Find its area.
8. **(Proof track.)** Give a complete justification. Find the distance between the parallel lines $3x+4y=5$ and $3x+4y=30$ .
9. **(Proof track.)** Give a complete justification. A circle centered on $x=2$ passes through $(0,1),(4,5)$ . Find its radius.