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Understanding Rectangles, Rhombi, and Squares: Properties and Examples

This guide explores the properties of rectangles, rhombi, and squares, essential quadrilaterals in geometry. Rectangles feature four right angles, congruent opposite sides, and bisecting congruent diagonals. The guide provides examples for finding unknown variables in rectangle dimensions. It also discusses rhombi, characterized by four congruent sides and perpendicular diagonals, and squares, which combine all properties of both rectangles and rhombi. Learn how to verify these shapes using coordinates and slopes on the coordinate plane.

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Understanding Rectangles, Rhombi, and Squares: Properties and Examples

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  1. 8-4 Rectangles

  2. Properties of Rectangles • Quadrilateral with 4 right angles • Opposite sides are congruent • Opposite angles are congruent • Consecutive angles are supplementary • Diagonals are congruent and bisect each other • Parallelogram

  3. Examples: • RSTU is a rectangle. If RT = 6x+4 and SU = 7x-4, find x • LMNP is a rectangle, find x and y S T R U 6y+2 5x+8 3x+2

  4. Diagonals • If the diagonals of a parallelogram are congruent then the parallelogram is a rectangle. • Ex: A contractor has been hired to pour a concrete patio. How can he be sure that the frame in which to pour the concrete is rectangular?

  5. On the coordinate plane • Find the slope of all sides • If the 2 consecutive sides have slopes that are opposite reciprocals then the sides are perpendicular (form right angles) • Opposite reciprocals

  6. Example: Is ABCD a rectangle if A(-2, 1) B(4, 3) C(5, 0) and D(-1, -2)

  7. 8-5 Rhombi and Squares

  8. Properties of Rhombi • Quadrilateral with 4 congruent sides • Diagonals are perpendicular • Each diagonal bisects a pair of opposite angles • If the diagonals of a parallelogram are perpendicular then the parallelogram is a rhombus • Has all the properties of a parallelogram

  9. Examples: • LMNP is a rhombus, Find y and m<PNL M 1 L N Q P m<1 = y2 – 54 m<MLP = 64

  10. Properties of a Square • Has all the properties of a parallelogram • Has all the properties of a rhombus • Has all the properties of a rectangle

  11. Square Rect. with 4 Congruent sides Rhombus Parallelogram With 4 congruent sides Rectangle Parallelogram With 1 rt < Parallelogram Opposite sides parallel

  12. Example: • Determine whether ABCD is a rhombus, a rectangle or a square for A(-2, -1) B(-1, 3) C(3, 2) and D(2, -2)

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