1. In a holodeck simulation, the Starship Enterprise at point A is traveling to meet the Excelsior Transport Ship at point B. The ship at point B is transporting an alien ambassador who must board the Enterprise but cannot use the transporter. The ships have agreed to meet at the midpoint of the line segment connecting the two ships’ current positions. a) Find the holodeck grid coordinates of the meeting location? Which of the formulas above is useful?

b) Find the holodeck distance between the ships’ current positions, to the nearest hundredth of a grid unit.

c) The distance found in part (b) is measured in grid units, not miles. If each grid unit in the holodeck represents 400 actual miles, find the distance to the nearest mile between the ships’ current positions. Use your rounded answer from (b).

The Pythagorean Theorem states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the two remaining sides. It is important to remember that the theorem applies only to right triangles. It is also helpful to note that c (the hypotenuse) must be the longest side.
Cutting down a tree:
2. You need to remove a tree from your front yard. You have climbed the tree and trimmed off the branches, and you are now ready to chop down the trunk. It is very important that the tree falls in the right direction… away from your house. You have attached a wire to the top of the trunk. You will attach the other end of the wire to the ground and pull it tightly so that when the trunk begins to fall, the wire will pull it in the right direction. The trunk is 24 feet tall and you are going to attach the wire to the ground 10 feet away from the base of the trunk.
a) How long must your wire be to reach the ground?

b) What assumption about the tree and the ground did we need to make here? Why did we need to make it?
(Hint: What formula did you use, and what has to be true in order for you to be able to use it?)

Measuring the depth of a lake:
3. You need to measure the depth of a large lake. Since the sonar equipment is very expensive, you decide to use your friend’s boat. The boat has an anchor on a 100 ft line. You take the boat out to the middle of the lake on a windy day. You drop the anchor and let the wind push the boat until the anchor line is tight. Your GPS tells you that the boat has moved 82 feet. Assuming the bottom of the lake is flat, what is the depth of the lake?

Simplify your answer (this is an EXACT answer, so breakdown your radicand and bring any numbers outside the radical that you can), then, round (approximate) it to the nearest foot.

Is the foundation square?
4. You have hired a contractor to build a house. The rectangular foundation (the concrete slab that the house is built on) has just been finished. You want to make sure that the construction is done well and the foundation is square, so you decide to see if the corners are right angles. You measure the width of the foundation to be 30 feet, the length to be 72 feet and the diagonal to be 78 feet.

Did the company build the foundation with right angles? Prove your answer by showing work.