Analytic Geometry Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Determine whether points P(1,2)P(1,2), Q(3,6)Q(3,6), and R(5,10)R(5,10) are collinear using the slope method.

Solution

  1. 1
    Slope of PQPQ: mPQ=6−23−1=42=2m_{PQ} = \frac{6-2}{3-1} = \frac{4}{2} = 2.
  2. 2
    Slope of QRQR: mQR=10−65−3=42=2m_{QR} = \frac{10-6}{5-3} = \frac{4}{2} = 2.
  3. 3
    Since the slopes are equal and the segments share point QQ, all three points lie on the same line.

Answer

Yes, PP, QQ, RR are collinear (all on the line y=2xy = 2x).
Three points are collinear if any two consecutive pairs have the same slope. Equal slopes with a shared point confirm they all lie on a single line.

About Analytic Geometry

Analytic geometry studies geometric objects using coordinate systems and algebraic equations, translating shapes into formulas so that algebra can solve geometry problems. This field, founded by Descartes, unifies algebra and geometry.

Learn more about Analytic Geometry →

More Analytic Geometry Examples

Example 1 medium

Prove that the triangle with vertices [formula], [formula], and [formula] is equilateral using coord

Example 2 hard

Use coordinates to prove that the diagonals of a rectangle bisect each other.

Example 4 medium

Show that quadrilateral [formula], [formula], [formula], [formula] is a parallelogram by comparing s

← All Analytic Geometry Examples Analytic Geometry Concept