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Kite Design Sara would like to develop her own kite. She was told that she was to make sure that the inside angle of the tail is 60. What would the other angles of her kite be to the nearest degree if she makes the kite 40 inches from head to tail and 30 inches from side to side? If you want a kite that is twice the area of Sara’s and all other criteria stays the same, what are the horizontal and vertical lengths of the new kite? Teacher Notes: Materials: Pencil, paper, calculator Standards Addressed: 0, 1, 2, 5 GLEs: 1.2.1, 1.3.2, 1.3.4, 1.5.6 Scaffolding Questions: 1. 2. 3. 4. 5. 6. What are the characteristics of a kite shape? What is the relationship of the diagonals? What are the properties of right triangles? What might be a trigonometric function that’s useful here? How do we find the area of a triangle? How does changing the dimensions of a figure affect its area? Sample Solution #1: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Draw the kite figure. Use trigonometric functions to find the height of a lower triangle. OR: Use the properties of a special right triangle to find the height of a lower triangle. Use subtraction to find the height of the upper triangle. Use trigonometric functions to find the upper angles. (This is the answer to the first question.) Use the area of a triangle formula to find the area of the upper and lower triangles; double for the area of the kite. Draw the larger kite; fill in the same angles. Double the area of the lower triangle. Use the formula for area and properties of special right triangles (x3 is the height if x is the base) to get the length of the horizontal diagonal. Use x3 to get the height of the lower triangle. Use the inverse trigonometric function to find the height of the upper triangle. Add the answer to 8 and 9 to get the vertical diagonal. Extension Questions: 1. If you want a kite that is half the area of Sara’s kite and all the other criteria stay the same, what are the horizontal and vertical lengths of the new kite? Solution: Using proportions: the diagonals of the larger kite are 1.414 larger than the diagonals of Sara’s kite. Therefore, the diagonals of the smaller kite will be the diagonals of Sarah’s kite divided by 1.414. Or: Sarah’s dimensions times 2 is equal to the dimensions of the larger kite. And, Sarah’s dimensions divided by 2 is equal to the dimensions of the smaller kite. 2. What is the relationship of the areas of the kite half the size of the original and the area of the kite twice the size of the original? Solution: The area of the largest kite is 4 times the area of the smallest kite, and the diagonals of the largest kite are twice the diagonals of the smallest kite.