Quadrilaterals Exploration

From EscherMath
Jump to navigationJump to search


Time-40.svg

Objective:

  • Familiarize yourself with the different classes of quadrilaterals.
  • Think critically about what exactly a definition tells us.
  • Create good, informative examples as well as counter-examples

Polygons, like living things, can be organized by a hierarchy of classes. Some members of the quadrilateral class are subclasses of other types of quadrilaterals. Squares, for example, are a subclass of rectangles, because a square is a rectangle whose sides are all the same length. Rectangles, however, are not a subclass of squares, because not all rectangles are squares.

One quadrilateral may be a subclass of more than one other kind of quadrilateral. Squares are a subclass of both rhombuses and rectangles. But since most rhombuses are not rectangles, and most rectangles are not rhombuses, neither is a subclass of the other. Below, read about a dozen types of quadrilaterals by definition, all have four sides, but each has unique qualities. Then answer the questions below.

  • Cv: Convex quadrilateral whose angles are all less than 180 degrees.
  • Cy: Cyclic quadrilateral: a quadrilateral whose corners all lie on a single circle.
  • K: Kite: a quadrilateral with two pairs of adjoining sides of equal length.
  • P: Parallelogram: a quadrilateral whose opposite sides are parallel.
  • Q: Quadrilateral: a polygon with four sides.
  • Rc: Rectangle: a quadrilateral with four 90° angles.
  • Rh: Rhombus: a quadrilateral whose sides are all the same length.
  • S: Square: a quadrilateral with four sides of equal length and four 90° angles.
  • T: Trapezoid: a quadrilateral having exactly two parallel sides.
  • U1: Unnamed quadrilateral 1: has opposite angles of equal measure.
  • U2: Unnamed quadrilateral 2: has at least one pair of opposite angles both 90°.
  • U3: Unnamed quadrilateral 3: has at least one pair of opposite sides of equal length.


Questions

  1. Draw representative illustrations (several) of these types of quadrilaterals.
  2. Which two definitions describe exactly the same type of quadrilateral?
  3. Besides Q, which two types of quadrilaterals are not necessarily convex?
  4. Quadrilaterals of class U2 (one pair of opposite angles both 90°) are a subclass of which two other types of quadrilaterals, not counting Q?
  5. Can you find a chain of five of the quadrilaterals in which each type is a subclass of the next? Hint: The first is S and the last is Q.
  6. Can you find one set of four quadrilaterals in which no one quadrilateral is a subclass of another? Hint: Try drawing pictures.

BONUS QUESTION: Can you find--or define--a fifth type of quadrilateral that is neither a subclass nor a super-class of these four quadrilaterals?

Handin: A sheet with answers to all questions.

Some portions originally from [1] and are Copyright 2000 The Walt Disney Company, used with permission.

  1. Scott Kim, Discover "Bogglers", July 2000