3.2+Puzzling+Through+a+Proof

10-28-08 SMJ


 * __3.2 Puzzling Through a Proof__**

Big Idea: The dimensions of a right triangle can be determined with limited information. Essential Question: How can I find the perimeter of a right triangle? Notes from Class: No notes were taken


 * __Problem 3.2__**

The length of shortest leg of triangular piece is equal to the side length of smallest square, the length of longer leg is equal to the side length of medium square, and the length of hypothenuse is equal to the side length of the biggest square.
 * A. Cut out the puzzle pieces. Examine a triangular piece and the three square pieces. How do the side lengths of the squares compare to side lengths of the triangle?**

Look at the picture below.( T is triangular piece, BS is biggest square, MS is medium square, and SS is smallest square) I conclude that small and medium squares' areas are equal to big square's area. Which prooves a^2(SS area)+b^2(MS area)=c^2(BS area). Because if you take out the triangular pieces, since there are 4 in each frames, then you get 1 BS on the first frame and MS and SS on the other.
 * B. Arrange the 11 puzzle pieces to fit exactly into the two puzzle frames. Use four triangles in each frame.**
 * C. Carefully study the arrangements in the two frames. What conclusion can you draw about the relationship among the areas of the three square puzzle pieces**__?__

It means small leg^2(area of SS)+medium leg^2(area of MS)=hypothenuse^2(area of BS). Or a^2+b^2=c^2
 * D. What does the conclusion you reached in part C mean in terms of the side lengths of the triangles?**

a^2+b^2=c^2
 * __Problem 3.2 Follow-Up__**
 * 1. In Problems 3.1 and 3.2, you explored the Pythagorean Theorem. State this relationship as a general rule for any right triangle with legs of lengths //a// and //b// and a hypothenuse of length //c//.**

2a. a^2+b^2=c^2 -> 3^2+5^2=c^2 -> 9+25=34. The area of a square is 34 cm.sq. 2b. The length of the hypothenuse is sqrt of 34.
 * 2. A right triangle has legs of lengths 3 cm and 5 cm.**
 * a. Use the Pythagorean Theorem to find the area of a square drawn on teh hypothenuse of the triangle.**
 * b. What is the length of the hypothenuse?**

3a. The area of a square is 169 in.sq. 3b. The length of the hypothenuse is 13 in.
 * 3. A right triangle has legs of lengths 5 in and 12 in.**
 * a. Find the area of a square drawn on the hypothenuse of the triangle.**
 * b. What is the length of the hypothenuse?**

a^2+b^2=c^2 -> 3^2+6^2=c^2 -> 9+36=45. Length of the hypothenuse is sqrt of 45 cm.
 * 4. Use the Pythagorean Theorem to find the length of the hypothenuse of this triangle. (Has leg of 3 and 6 cm.)**

c^2-a^2=b^2 -> 15^2-9^2=b^2 -> 225-81=144. Sqrt of 144 is 12. The other leg is 12 cm long.
 * 5. the hypothenuse of a right triangle is 15 cm long, and one leg is 9 cm long. How long is the other leg**?