Exercise 19 Page 417

We are asked to prove that the midpoints of the sides of a rectangle form a rhombus. We are asked to write a coordinate proof, so let's place the rectangular flag in a coordinate plane so that two of its sides are along the coordinate axes.

Before deciding on the expressions for the coordinates of the vertices of rectangle ABCD, let's recall some properties.

• Side BC is parallel to the x-axis, so the y-coordinates of B and C are the same.
• Side CD is parallel to the y-axis, so the x-coordinates of C and D are the same.

  Since we also need the coordinates of the midpoints of the sides, we will use B(0,2b), C(2d,2b), and D(2d,0) for the vertices.

  

  On vertical segments, the midpoint has the same x-coordinate as the two endpoints and the y-coordinate is the average.

  • The midpoint between points A(0,0) and B(0,2b) is M_(AB)(0,b).
  • The midpoint between points C(2d,2b) and D(2d,0) is M_(CD)(2d,b).

  On horizontal segments, the midpoint has the same y-coordinate as the two endpoints and the x-coordinate is the average.

  • The midpoint between points D(2d,0) and A(0,0) is M_(DA)(d,0).
  • The midpoint between points B(0,2b) and C(2d,2b) is M_(BC)(d,2b).
  

  We can again use our observations above to find the midpoints of the diagonals.

  • The midpoint between points M_(AB)(0,b) and M_(CD)(2d,b) is (d,b).
  • The midpoint between points M_(BC)(d,2b) and M_(DA)(d,0) is (d,b).

  Let's collect what we know about diagonals M_(AB)M_(CD) and M_(BC)M_(DA)

  Claim	Justification
  M_(AB)M_(CD) is horizontal.	The endpoints M_(AB)(0,b) and M_(CD)(d,b) have the same y-coordinates.
  M_(BC)M_(DA) is vertical.	The endpoints M_(BC)(d,2b) and M_(DA)(d,0) have the same x-coordinates.
  The diagonals are perpendicular.	Vertical and horizontal lines are perpendicular.
  The diagonals bisect each other.	Point (d,b) is the midpoint of both diagonals.

  Let's finish the proof by recalling two theorems.

  • Since the diagonals bisect each other, according to Theorem 6-11, the quadrilateral formed by the midpoints is a parallelogram.
  • Since the diagonals are perpendicular, according to Theorem 6-16, the parallelogram is a rhombus.