M20 - Professor: Jennifer Plants PDF

Preview

Summary

Professor: Jennifer Plants...

Description

Two out of three. If a right triangle has legs of length 1 and 2, what is the length of the hypotenuse? Enter the exact answer. If it has one leg of length 1 and a hypotenuse of length 9, what is the length of the other leg? Enter the exact answer. √5 √80 In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Hypotenuse hype. If a right triangle has legs of length 1 and x, what is the length of the hypotenuse? Enter the exact answer. √x to the second power + 1 Assessing area. Suppose you know the base of a rectangle has a length of 5 inches and a diagonal has a length of 13 inches. Find the area of the rectangle. 60 We know the base of the rectangle; we need to find the height. The diagonal divides the rectangle into two right triangles, each with one leg of length 5 and hypotenuse of length 13. The length of the other leg will be the height of the rectangle. If x denotes the length of this leg, the Pythagorean Theorem tells us that 132 = x2 + 52. So x2 = 169 - 25 = 144, yielding x = 12. Thus the area of the rectangle is 5 × 12 = 60 square inches. Operating on the triangle. Using a straightedge, draw a random triangle. Now, carefully cut it out. Next, amputate the angles by snipping through adjacent sides. Now, move the angles together so the vertices all touch and the edges meet. 180 degrees When the angles are aligned, they will form a straight line or half of a complete rotation. Because a full rotation corresponds to 360 degrees, we have that the sum of the angles in a triangle is 180 degrees. Easy as 1, 2, 3? Can there be a right triangle with sides of length 1, 2, and 03? Can you find a right triangle whose side lengths are consecutive natural numbers? The triangle with sides: No 3-4-5

In any right triangle, the side opposite the right angle is the largest, so in our case, the length of the hypotenuse is 3. If this is indeed a right triangle, then the Pythagorean Theorem must hold, that is, 12 + 22 = 32, but 5 isn't 9, and so we don't have a right triangle. The only set of consecutive numbers that works here is 32 + 42 = 52. Getting a pole on a bus. For his 13th birthday, Adam was allowed to travel down to Sarah's Sporting Goods store to purchase a brand new fishing pole. With great excitement and anticipation, Adam boarded the bus on his own and arrived at Sarah's store. Although the collection of fishing poles was tremendous, there was only one pole for Adam and he bought it: a five-foot, one-piece fiberglass "Trout Troller 570" fishing pole. When Adam's bus arrived, the driver reported that Adam could not board the bus with the fishing pole. Objects longer than four feet were not allowed on the bus. In tears, Adam remained at the bus stop holding his beautiful five-foot Trout Troller. Sarah, seeing the whole ordeal, rushed out and said, "Don't cry, Adam! We'll get your fishing pole on the bus!" Sure enough, when the same bus and the same driver returned, Adam boarded the bus with his fishing pole and the driver welcomed him aboard with a smile. How was Sarah able to have Adam board the bus with his five-foot fishing pole without breaking the bus line rules and without cutting or bending the pole? Sarah somehow found a 3 foot by 04 foot box and lay Adam's fishing pole along the diagonal. When we measure the length of a box, we don't measure the length of the diagonal, we measure the lengths of the sides—even though the diagonal is larger than either of the sides! So Sarah somehow found a 3 foot by 4 foot box and lay Adam's fishing pole along the diagonal. Once again, the 3-4-5 triplet comes to the rescue. The scarecrow. In the 1939 movie The Wizard of Oz, when the brainless scarecrow is given the confidence to think by the Wizard (by merely handing him a diploma, by the way), the first words the scarecrow utters are, "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side." An isosceles right triangle is just a right triangle having both legs the same length. Suppose that an isosceles right triangle has legs each of length 2. What is the length of the hypotenuse? Do not round. Is the scarecrow's assertion valid? This question illustrates the true value of a diploma without studying. 2*2 to the 1/2 power (or 3√2?) No Rooting through a spiral. Start with a right triangle with both legs having length 01. What is the length of the hypotenuse? Do not round. Suppose we draw a line of length 01 perpendicular to the hypotenuse and then make a new triangle as illustrated (see figure below). What is the length of this new hypotenuse?

Suppose we continue in this manner. Describe a formula for the lengths of all the hypotenuses, i.e. find the length of the hypotenuse of the Nth triangle. √2 √3 (n + 1) to the second power Getting squared away. In our proof of the Pythagorean Theorem, we stated that the second figure is actually two perfect squares touching along an edge. Can you prove that they are indeed both perfect squares? Yes, it can be proved. Before moving the top triangle, side a represents the length of one side of the alleged square. After the triangle is moved, the same side represents the other side of the alleged square. This proves that the figure is indeed a perfect square. The second square is proved in a similar manner. Standing guard. Draw the floor plan of a gallery with three vertices. What shape do you get? What is the smallest number of guards you need? Triangle 1. A gallery with three sides will have a floor plan in the shape of a triangle requiring only one guard....