Proofs involving quadrilaterals I

Learning Objectives Identify the key properties of parallelograms, rectangles, rhombuses, and squares. Apply the distance formula to determine and compare the side lengths of quadrilaterals on a coordinate plane. Apply the slope formula to determine if sides of a quadrilateral are parallel or perpendicular. Apply the midpoint formula to determine if the diagonals of a quadrilateral bisect each other. Construct a coordinate proof to classify a quadrilateral as a parallelogram. Construct a coordinate proof to classify a parallelogram as a rectangle, rhombus, or square. Ever wonder how a builder ensures a window frame is a perfect rectangle and not just a lopsided parallelogram? It's all about proof! 📐 This tutorial will introduce you to the world of geometric proofs for...

TermDefinitionExample ParallelogramA quadrilateral where both pairs of opposite sides are parallel.Given quadrilateral ABCD, if AB is parallel to DC and AD is parallel to BC, then ABCD is a parallelogram. RectangleA parallelogram with four right angles. This also means its diagonals are congruent.A parallelogram ABCD is a rectangle if angle A is 90°, or if the length of diagonal AC is equal to the length of diagonal BD. RhombusA parallelogram with four congruent (equal length) sides. This also means its diagonals are perpendicular.A parallelogram ABCD is a rhombus if AB = BC = CD = DA. SquareA parallelogram that is both a rectangle and a rhombus. It has four congruent sides and four right angles.A quadrilateral that has four equal sides and diagonals that are equal in length is a square....

Distance Formula d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} Use this to find the length of a side or diagonal. To prove sides are congruent, show their lengths are equal. To prove diagonals are congruent, show their lengths are equal. Slope Formula m = \frac{y_2 - y_1}{x_2 - x_1} Use this to check for parallel or perpendicular lines. Parallel lines have equal slopes (m₁ = m₂). Perpendicular lines have negative reciprocal slopes (m₁ * m₂ = -1). Midpoint Formula M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) Use this to find the midpoint of a segment. The most common use is to prove that diagonals bisect each other by showing they share the same midpoint.