Exercises

1. 1. On each quadrilateral, draw the diagonals and indicate whether the diagonals bisect each other, are perpendicular to each other, and whether the diagonals are equal in length.

      Square
      

      Trapezoid
      

      Rectangle

      

      Rhombus
      

      Parallelogram
      

   2. List all of the quadrilaterals from part a) that fit each description.
      1.   have diagonals that always bisect each other.
      2.   have diagonals that are always perpendicular to each other.
      3.   have diagonals that are always equal in length.
2. How are the diagonals of a square similar to the diagonals of a rhombus?  How are they different?
3. You are shown just the diagonals of a quadrilateral and are asked if the shape would be best described as a trapezoid or parallelogram.  How could you determine which type of quadrilateral you have?
4. Yukiko has a rectangle and notices that the diagonals are perpendicular to each other.  Can Yukiko more specifically classify this shape?
5. Lilliana is helping her Mom build a shed in the backyard.  One wall of the shed will measure \(3.6\) m by \(2.4\) m. When they build the frame for the wall, how could they use a measuring tape and their knowledge of diagonals to confirm that the wall is rectangular and not slanted?
6. The parallelogram, \(ABCD\), has a base of \(4\) cm and a height of \(2\) cm.  Another copy of this parallelogram is rotated \(180\) degrees and placed directly on top as shown in the diagram.  The vertices, \(CDEF\), create a new shape.  What can be said about the diagonals of the new shape?
   
7. Shara chooses a mystery quadrilateral that is either a trapezoid, a parallelogram, a rhombus, a rectangle, or a square.  She tells you that the quadrilateral has diagonals that are not equal in length.
   1. Based on this information, what different quadrilateral types are possible?
   2. Its diagonals are also not perpendicular to each other.  What different quadrilateral types are now possible?
   3. Its diagonals also do not bisect each other.  What type of quadrilateral must Shara's mystery quadrilateral be?
8. A cyclic polygon is a polygon for which all of its vertices lie on a single circle.  It can be shown that all triangles are cyclic, but only some quadrilaterals are cyclic.
   Cyclic
   
   

   Non-cyclic
   
   To determine if a quadrilateral is cyclic, it is not enough to know the lengths of the four sides of the quadrilateral.  However, it is enough to know the lengths of the sides and the lengths of the two diagonals of the quadrilateral.  
   1. Can you explain why every triangle is cyclic?  Can you explain why not every quadrilateral is cyclic?
   2. Can you explain why knowing the lengths of the sides of a quadrilateral is not enough to classify the quadrilateral as cyclic or non-cyclic?  Hint: Compare a rectangle and parallelogram that have identical side lengths.
   3. Ptolemy's Theorem states that a quadrilateral \(ABCD\), with side lengths \(AB = a\), \(BC = b\), \(CD = c\), and \(AD = d\) and diagonals \(AC = f\) and \(BD = g\), is cyclic exactly when the following relationship is satisfied:\[a \times c + b \times d = f\times g\]
      

      Using Ptolemy's Theorem, determine whether or not the following quadrilaterals are cyclic:

      1. 
      2. 
      3. 
   4. Given that a unit square is a cyclic quadrilateral, use Ptolemy's Theorem to show that the length of the diagonal of a unit square must be \(\sqrt{2}\).