⭕ Circles & Analytic Geometry
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Coordinate Geometry Proofs

Prove geometric theorems using coordinates and algebraic methods.

Coordinate geometry uses algebraic tools — the distance formula, midpoint formula, and slope — to prove geometric theorems. Placing figures strategically on the coordinate plane simplifies proofs significantly.

Distance: d=(x2−x1)2+(y2−y1)2\text{Distance: } d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}
Midpoint: M=(x1+x22,  y1+y22)\text{Midpoint: } M = \left(\frac{x_1+x_2}{2},\; \frac{y_1+y_2}{2}\right)

Key Strategies for Coordinate Proofs

Place one vertex at the origin to simplify coordinates.Align sides along axes when possible.Use general coordinates like (a, 0), (0, b) to keep proofs general.Parallel lines: equal slopes. Perpendicular: slopes are negative reciprocals.
🔍Proving a Quadrilateral is a Parallelogram
Given A(0,0), B(4,0), C(6,3), D(2,3). Show opposite sides are parallel: slope AB = 0, slope DC = 0 (parallel). Slope BC = 3/2, slope AD = 3/2 (parallel). It is a parallelogram.
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Remember This!

To prove a quadrilateral is a rectangle, show it is a parallelogram AND that one angle is 90° (adjacent sides have perpendicular slopes).

✏ī¸ Try It!

What formula proves two segments bisect each other (share the same midpoint)?

Practice with CalcVerse
đŸ—ēī¸Distance & Midpoint
Take Quiz 📝 — 24 Questions