Student Conjectures: Two friends are trying to decide how long their ladder should be for the zip line they are building. 1. What is each person suggesting? (1 point) Student Conjecture Will Thomas What do you think? 2. Which ladder length would you pick? Why? (1 point) What problems would you encounter if the ladder were too short or too long? (1 point) Draw a Diagram: 3. What key details are given in the scenario? (1 point) 4. Sketch a diagram for the situation. Clearly label which ladder length you are using. (Assume the other end of the zip line is anchored to the ground.) (1 point) 5. Label the angles and side lengths of your sketch. (1 point) 6. In the space below, use the law of sines to find: (4 points total: 1 point each) a. The length of the zip line b. The height of the point on the tree where the top of the ladder rests against it c. The distance between the base of the ladder and the base of the tree d. The distance between the base of the tree and the spot where the zip line is anchored to the ground Make a Comparison: 7. Now let's sketch a diagram for the other ladder length. (1 point) 8. Label the interior angles of your sketch. (1 point) 9. Use the law of sines to find: (4 points: 1 point each) a. The length of the zip line b. The height of the point on the tree where the top of the ladder rests against it c. The distance between the base of the ladder and the base of the tree d. The distance between the base of the tree and the spot where the zip line is anchored in the ground Make a Decision: 10. Based on the calculations you did using the law of sines, did you make a good choice for which ladder to use? Why or why not? (1 point) Verify Your Calculations: How can we be sure the law of sines is correct? Let's check your answers using a trigonometric relationship you are very familiar with: the Pythagorean theorem. Assuming that the tree is perpendicular to the ground, it can be treated as the shared leg of two right triangles. Does your diagram show this? 11. How can the Pythagorean theorem prove that the tree really is part of two different right triangles? (1 point) 12. Using the side lengths you found above, apply the Pythagorean theorem to confirm or reject our hypothesis that the tree forms two right triangles. (1 point) 13. How convincing is this proof? Explain your reasoning. (1 point)