Analytic Geometry Math Example 4

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

Example 4

medium
Show that quadrilateral A(1,1)A(1,1), B(4,2)B(4,2), C(5,5)C(5,5), D(2,4)D(2,4) is a parallelogram by comparing slopes of opposite sides.

Solution

  1. 1
    Slope of ABAB: 2−14−1=13\frac{2-1}{4-1} = \frac{1}{3}. Slope of DCDC: 5−45−2=13\frac{5-4}{5-2} = \frac{1}{3}. So AB∥DCAB \parallel DC.
  2. 2
    Slope of BCBC: 5−25−4=3\frac{5-2}{5-4} = 3. Slope of ADAD: 4−12−1=3\frac{4-1}{2-1} = 3. So BC∥ADBC \parallel AD.
  3. 3
    Both pairs of opposite sides are parallel, so ABCDABCD is a parallelogram.

Answer

ABCDABCD is a parallelogram since both pairs of opposite sides are parallel.
A quadrilateral is a parallelogram if and only if both pairs of opposite sides are parallel. Parallel lines have equal slopes, making this straightforward to verify with coordinates.

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 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 3 easy

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

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