Analytic Geometry

Coordinate strategy

Choose axes and place points so the geometry becomes simple. Put a convenient vertex at the origin, align a side with an axis, or use symmetry when allowed.

A smart coordinate choice reduces algebra without changing the geometric relationships.

Lines as equations

A nonvertical line can be written $y=mx+b$, while a vertical line is $x=c$. Slope measures direction; intercepts locate axis crossings.

Two line equations can be solved as a system to find their intersection.

Worked example: prove a triangle is right

The coordinates make both the proof and measurement direct.

Transformations

Coordinate rules represent translations, reflections, rotations, and dilations. Rigid transformations preserve length and angle; dilations preserve shape and angle but scale lengths.

Composition applies several transformations in sequence, and order can matter.

Common mistakes

Choosing awkward coordinates without using symmetry. Simplify placement when permitted.

Assuming a diagram proves parallel or perpendicular lines. Calculate slopes.

Using distance where squared distance would suffice and introducing rounding. Keep exact forms.

Proving only one property of a specialized shape. Meet the definition completely.

Keeping extraneous intersection roots. Verify candidates.

Quick self-check

• Is the coordinate system chosen to simplify the geometry?
• Which equations represent the lines or curves?
• Are slope, distance, and midpoint used for the right claims?
• Does a shape proof meet all required defining conditions?
• Are loci and intersections interpreted as geometric sets?
• Are exact values, units, and contextual domains preserved?