5. How do the base and height of the parallelogram compare to the base and height of the original triangle? Write an expression for the height of the parallelogram in terms of the height h of the triangle.
6. Write your formula for the area of a parallelogram from Activity 1. Substitute the expression you wrote for the height of the parallelogram into this formula. You now have a formula for the area of a triangle.

Activity 3

• Step 1 Count and record the bases and height of the trapezoid below.
• Step 2 Copy the trapezoid. Mark the midpoints M and N, and draw midsegment M N ¯ . m n bar , .
• Step 3 Cut out the trapezoid. Then cut it along M N ¯ . m n bar , .
• Step 4 Transform the trapezoid into a parallelogram.

7. What transformation did you apply to form a parallelogram?
8. What is an expression for the base of the parallelogram in terms of the two bases, b 1 b sub 1  and b 2 , b sub 2 , comma  of the trapezoid?
9. If h represents the height of the trapezoid, what is an expression in terms of h for the height of the parallelogram?
10. Substitute your expressions from Questions 8 and 9 into your area formula for a parallelogram. What is the formula for the area of a trapezoid?

Exercises

11. In Activity 2, can a different rotation of the small triangle form a parallelogram? If so, does using that rotation change your results? Explain.
12. Make another copy of the Activity 2 triangle. Find a rotation of the entire triangle so that the preimage and image together form a parallelogram. How can you use the parallelogram and your formula for the area of a parallelogram to find the formula for the area of a triangle?
13. 1. In the trapezoid below, a cut is shown from the midpoint of one leg to a vertex. What transformation can you apply to the top piece to form a triangle from the trapezoid?
    2. Use your formula for the area of a triangle to find a formula for the area of a trapezoid.

    
    Image Long Description

14. Count and record the lengths of the diagonals, d 1 d sub 1  and d 2 , d sub 2 , comma  of the kite above. Copy and cut out the kite. Reflect half of the kite across the line of symmetry d 1 d sub 1  by folding the kite along d 1 . d sub 1 , .  Use your formula for the area of a triangle to find a formula for the area of a kite.

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