Coordinate Geometry (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

Coordinate Geometry: core idea

Circles/lines.

Find the distance between $(-2,1)$ and $(4,9)$ .

Hint: Translate the geometry into coordinates before simplifying. Answer: 10.

1. Find the area of the triangle with vertices $(0,0),(6,0),(2,5)$ .
2. Lines $y=2x+1$ and $y=-x+7$ intersect. Find the $x$ -coordinate.
3. Find the equation's $y$ -intercept: $3x+4y=20$ .
4. A circle centered at $(2,-1)$ passes through $(5,3)$ . Find its radius.
5. Find the slope of the line through $(2,7)$ parallel to $3x+2y=4$ .
6. Find the area of the polygon with vertices $(0,0),(4,0),(4,3),(0,3)$ .
7. The midpoint of $(x,4)$ and $(6,10)$ is $(4,7)$ . Find $x$ .
8. Find the distance from $(3,4)$ to the origin after rotating it $90^\circ$ about the origin.
9. A line through $(1,2)$ has slope $3$ and meets the $x$ -axis at $(a,0)$ . Find $a$ .