Analytic Geometry Math Example 1

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

Example 1

medium
Prove that the triangle with vertices A(0,0)A(0,0), B(4,0)B(4,0), and C(2,23)C(2,2\sqrt{3}) is equilateral using coordinates.

Solution

  1. 1
    Compute ABAB: AB=(4−0)2+(0−0)2=16=4AB = \sqrt{(4-0)^2 + (0-0)^2} = \sqrt{16} = 4.
  2. 2
    Compute BCBC: BC=(2−4)2+(23−0)2=4+12=16=4BC = \sqrt{(2-4)^2 + (2\sqrt{3}-0)^2} = \sqrt{4 + 12} = \sqrt{16} = 4.
  3. 3
    Compute CACA: CA=(0−2)2+(0−23)2=4+12=4CA = \sqrt{(0-2)^2 + (0-2\sqrt{3})^2} = \sqrt{4+12} = 4.
  4. 4
    Since AB=BC=CA=4AB = BC = CA = 4, all three sides are equal, so the triangle is equilateral.

Answer

The triangle is equilateral with all sides equal to 44.
Analytic geometry lets us prove geometric properties by computing distances with the distance formula. Equal side lengths confirm an equilateral triangle without any angle measurements.

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.

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More Analytic Geometry Examples

Example 2 hard

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

Example 3 easy

Determine whether points [formula], [formula], and [formula] are collinear using the slope method.

Example 4 medium

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

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