must all be handwrit.ten and show all work and steps

0 Show your work for any calculations needed or simplifications made! . Clearly show your work and the steps necessary to solve each problem, including sketching diagrams that represent each situation. 1. A perfect triangle is a triangle with two key characteristics: 1) all three side lengths are positive integers, and 2) the area of the triangle area is equal to its perimeter. Is a triangle with sides of length 8, 9, and 16 a perfect triangle? Set up expressions and show the work needed to solve in order to answer the question. 2. Forest fire tower Laurel is located 90 miles due west of forest fire tower Springs. A forest fire is located on a bearing of N 44'J E from tower Laurel and a bearing of N 160 E from tower Springs (the eastern station). There is a fire-fighting crew located 38 miles from station Springs on a bearing of N 61 U E (from station Springs). a. Draw a diagram and label it with the given information to represent this situation. b. Determine the distance of the fire from both fire stations. 0. How far is the fire-fighting crew from the fire? 3. An Olympic-distance triathlon includes a 1.5 km swim. To determine the swim course of an Olympic-distance triathlon in Shoe Lake, the planners need to first find the distance between points A and B on opposite sides of a lake in order to determine where to place the buoys that will mark the swim course. You have been called in to help with determining the distances. From a point C on the shore, you find that CA = 259 m, 08 = 423 m, and angle ACB measures 132\". a. Determine the distance across the lake. b. Is the lake long enough to simply have an "out-and-back\" swim course for this triathlon (that is: swimming from point A to point B and then back to point A)? If not, what additional distance will need to be set off of the straight-line distance between A and Bin order to meet the distance requirements for this triathlon? Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false. (4 points each.) 4(4n + 1)(8n + 7) 5 1.4-6+5-7+6v8+...+4n(4n+2)= 2.12+42+72+...+(3n-2)2=M 2 For the given statement P n , write the statements P 1 , P k, and P m . (2 points) 3.2+4+6+...+2n=n(n+1)