Midpoint Madness Unit 2A: Geometric Proofs and Modeling: Proofs

Math 3 G.GPE.4,G.GPE.5,G.GPE.7 Assessment Title: Midpoint Madness Unit 2a: Geometric Proofs and Modeling: Proofs

1. Using Square ABCD, find the coordinates of the midpoints of each side.

2. Connect the midpoints to get the midpoint quadrilateral. What properties does this quadrilateral have, and how do you know?

3. Find the perimeter of Square ABCD and the perimeter of its midpoint quadrilateral. How do they compare?

4. Find the area of Square ABCD and the area of its midpoint quadrilateral. How do they compare?

5. If you were to find the next midpoint quadrilateral, how do you think its perimeter and area would compare to the previous one’s?

6. It appears that the midpoint quadrilateral of a square is another square. Do you think this is always true? Provide reasoning for your answer.

7. Do you believe the midpoint quadrilateral of any rectangle will always be a rectangle?

Let’s investigate with a 4 X 6 notecard Consider a 4 X 6 notecard and label it rectangle ABCD.

A D

B C

Find the midpoints of Sides AB, BC, CD, and DA. Call these points respectively EFGH. Find the midpoints of EF, FG, GH, and HE. Call these points respectively IJKL. Find the midpoints of IJ, JK, KL, and LI. Call these points MNPQ.

8. Your original notecard ABCD is a rectangle. What sort of quadrilateral is its midpoint quadrilateral EFGH? Use properties of quadrilaterals to support your answer.

9. What sort of quadrilateral is the next midpoint quadrilateral IJKL? Use properties of quadrilaterals to support your answer.

10. Place your notecard ABCD on the coordinate plane in a strategic way. Use coordinates to verify your classification of each of the midpoint quadrilaterals.

Quadrilateral EFGH:

Quadrilateral IJKL:

Quadrilateral MNPQ:

11. Without drawing the next four midpoint quadrilaterals, predict what shapes they would be.

12. Starting with Rectangle ABCD, find the area and perimeter of each subsequent midpoint quadrilateral and record your answers in the table below. Then, use pattern recognition and geometric thinking to complete the table. Don’t worry about naming rectangles after stage 3.

Stage Quadrilateral Area ( in2 ) Perimeter ( in ) 0 ABCD 24 20 1 EFGH 2 IJKL 3 MNPQ 4 5 6 If N is even: N If N is odd:

13. Summarize your conjectures. a. The midpoint quadrilateral of a square is a ______. b. The midpoint quadrilateral of a rectangle is a ______. c. The midpoint quadrilateral of a rhombus is a ______.

14. What do you think the midpoint point quadrilateral of any quadrilateral is? Draw several different quadrilaterals and construct their midpoint quadrilaterals. What shape does the midpoint quadrilateral appear to be? How could you prove this?

15. Suppose Quadrilateral ABCD lies in the coordinate plane. Let (,)xy be the coordinates of 11 vertex A, (,)xy the coordinates of B, (,)xy the coordinates of C, and (,)xy the 22 33 44 coordinates of D. Use coordinates to prove the observation you made in #14.

(Taken from Illustrative Mathematics: http://www.illustrativemathematics.org/illustrations/605)

16. Therefore, the midpoint quadrilateral of any quadrilateral is always a ______. Can you prove this without using coordinates?