Concept Byte: Quadrilaterals in Quadrilaterals

Use With Lesson 6-9

TECHNOLOGY

Activity

Construct

• Use geometry software to construct a quadrilateral ABCD.
• Construct the midpoint of each side of ABCD.

• Construct segments joining the midpoints, in order, to form quadrilateral EFGH.

  

Investigate

• Measure the lengths of the sides of EFGH and their slopes.
• Measure the angles of EFGH.

What kind of quadrilateral does EFGH appear to be?

Exercises

1. Manipulate quadrilateral ABCD.
   1. Make a conjecture about the quadrilateral with vertices that are the midpoints of the sides of a quadrilateral.
   2. Does your conjecture hold when ABCD is concave?
   3. Can you manipulate ABCD so that your conjecture doesn't hold?
2. Extend Draw the diagonals of ABCD.
   1. Describe EFGH when the diagonals are perpendicular.
   2. Describe EFGH when the diagonals are congruent.
   3. Describe EFGH when the diagonals are both perpendicular and congruent.

3. Construct the midpoints of EFGH and use them to construct quadrilateral IJKL. Construct the midpoints of IJKL and use them to construct quadrilateral MNOP. For MNOP and EFGH, compare the ratios of the lengths of the sides, perimeters, and areas. How are the sides of MNOP and EFGH related?

   

4. Writing In Exercise 1, you made a conjecture as to the type of quadrilateral EFGH appears to be. Prove your conjecture. Include in your proof the Triangle Midsegment Theorem, “If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and half its length.”
5. Describe the quadrilateral formed by joining the midpoints, in order, of the sides of each of the following. Justify each response.
   1. parallelogram
   2. rectangle
   3. rhombus
   4. square
   5. trapezoid
   6. isosceles trapezoid
   7. kite

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