Every day on Tim's walk to school he comes to a river. He has to walk far out of his way to get to a shallow crossing point in the river. Tim would like to construct a bridge across the river to shorten his walk to school. Tim finds that the point at which he crosses the river is 2 miles away and that he makes a 60 degree turn to get back on his path towards school. Given that the angle between the bridge and current path to school will be 70 degrees, can you help Tim figure out how long of a bridge he will need to construct?
Task 1: a. Can you estimate the length of the bridge? Justify your estimate. b. The slider tool allows you to change the angle at the point where Tim crosses the River. Can you use the slider tool to help you justify your estimate? If so, how? c. Is there a relationship between an interior angle of a triangle and the side opposite that angle? Explain. Task 2: Set the slider to 60 degrees to model Tim's situation. At school today, Tim decided to draw a diagram for himself. Uncheck the "River" box to see Tim's diagram. He remembers learning in class recently that he can draw in the height of a triangle to divide it into two right triangles. Check the box next to "Tim's Idea" to see the right triangles Tim formed. Tim also remembers using trigonometric ratios to find missing sides and angles of a triangle. a. How can Tim use this information to help him find the missing length of his triangle? b. Can you find the missing length of his triangle? How does this compare to your estimate? c. What does your answer represent? Task 3: Tim has decided to write a book to help others who may have the same problem. He wants to find out if his method can be used to find a way to relate the angles and sides of any non-right triangle. Check the box labeled "Tim's Book" to try and help Tim. a. Find two expressions for h using the two right triangles. b. Using your expressions from part (a), show that . c. Refer back to Task 1 part (c), was your answer sufficient? Can you now better describe the relationship? The relationship is called the Law of Sines. The Law of Sines says that for any ABC, the ratio of the sine of an angle to the length of the opposite side of that angle is constant. Therefore: , , and . The Law of Sines is a useful tool because it allows you to use the sine ratio to solve for measures of angles and lengths of sides of any triangle.