All Labs

Coordinate Geometry & Conics Lab

Use coordinate geometry to prove properties of polygons and explore conic sections interactively. Enter polygon vertices to verify rectangles, parallelograms, and rhombi with algebraic proofs. Graph circles, ellipses, parabolas, and hyperbolas with foci, vertices, and asymptotes. Classify any second-degree equation using the discriminant.

Guided Experiment: Coordinate Proof Techniques

How can you use slopes and distances to prove that a quadrilateral is a rectangle, parallelogram, or rhombus?

Write your hypothesis in the Lab Report panel, then click Next.

Coordinate Plane

Controls

Proof Results

Enter vertices and click "Analyze Polygon" to see results.

Data Table

(0 rows)
#TrialShape/ConicClassificationKey MeasureValueNotes
0 / 500
0 / 500
0 / 500

Reference Guide

Distance & Midpoint

The distance formula finds the length between two points in the plane.

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

The midpoint formula finds the point exactly halfway between two endpoints.

M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

Slope & Parallel/Perpendicular

The slope measures the steepness and direction of a line segment.

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

Parallel lines have equal slopes. Perpendicular lines have slopes whose product is -1.

m1⋅m2=−1  ⟹  perpendicularm_1 \cdot m_2 = -1 \implies \text{perpendicular}

Conic Sections Standard Forms

Each conic section has a standard form centered at (h, k).

Circle
(x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2
Ellipse
(x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
Hyperbola
(x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1
Parabola
(x−h)2=4p(y−k)(x-h)^2 = 4p(y-k)

Conic Classification (B² - 4AC)

Given a general second-degree equation, the discriminant determines the conic type.

Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
  • B2−4AC<0B^2 - 4AC < 0 and A=CA = C → Circle
  • B2−4AC<0B^2 - 4AC < 0 and A≠CA \ne C → Ellipse
  • B2−4AC=0B^2 - 4AC = 0 → Parabola
  • B2−4AC>0B^2 - 4AC > 0 → Hyperbola