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# Math and Humanity MATH 116

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This 7 page Class Notes was uploaded by Breana Ullrich on Monday October 19, 2015. The Class Notes belongs to MATH 116 at Radford University taught by William Case in Fall. Since its upload, it has received 13 views. For similar materials see /class/224682/math-116-radford-university in Mathematics (M) at Radford University.

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Date Created: 10/19/15

Section 71 The Pythagorean Theorem The origins of right triangle geometry can be traced back to 3000 BC in Ancient Egypt The Egyptians used special right triangles to survey land by measuring out 345 and 5 1213 right triangles to make right angles The Egyptians mostly understood right triangles in terms of ratios or what would now be referred to as Pythagorean Triples The Egyptians also had not developed a formula for the relationship between the sides of a right triangle At this time in history it is important to know that the Egyptians also had not developed the concept of a variable The Egyptians most studied speci c examples of right triangles For example the Egyptians use ropes to measure out distances to form right triangles that were in whole number ratios In the next illustration it is demonstrated how a 345 right triangle can be form using ropes to create a right angle 5 knots 3 knots 4 knots Using ropes that had knots that where equality spaced the Egyptians could measure out right angles by making a 345 right angle or other right triangles with the rope It wasn t until around 500 BC when a Greek mathematician name Pythagoras discovered that there was a formula that described the relationship between the sides of a right triangle This formula was known as the Pythagorean Theorem Pythagorean Theorem In a right triangle the sum of the squares of the two legs is equal to the square of the hypotenuse U c2 azb2 Example 1 Determine if the triangle measured out by ropes has out a right angle If you count the number of knots on each side of the triangle you get a ratio of 6810 10 knob 6 knob 8 knob Substituting these values into the Pythagorean Theorem using 10 as the hypotenuse and the other two sides as the legs you can determine if the triangle is a right triangle 02 azb2 102 62 82 10 3664 100100 Since the formula checks the triangle is a right triangle which gives us a right angle Here are some examples of how the Pythagorean Theorem can be use to nd the missing side of a right triangle Example 2 Suppose the two legs of a right triangle are 5 units and 12 units nd the length of the hypotenuse To find the solution substitute the value of the legs into the Pythagorean Theorem and solve for the hypotenuse Let a 5 and b 12 and solve for c c2 52 122 c2 25 144 c2 169 J7 J169 c 13 Example 3 Suppose that the hypotenuse of a right triangle is 26 units and one leg is 10 units nd the measure of the other leg To find the solution substitute the value of the leg and hypotenuse into the Pythagorean Theorem and solve for the missing leg Given a 10c 26 nd b 262 102 b2 676100b2 676 100100 100b2 576 b2 Jig 2 4 b 24 Applications of the Pythagorean Theorem The Pythagorean Theorem has several real life applications This is due to the fact that so many problems can be model or represented by a right triangle If this is the case then values can be assigned to the sides of the triangle and the unknown value can be found by solving for the missing side of the triangle Here are some examples of applications of right triangles and the Pythagorean Theorem Example 4 An empty lot is 120 ft by 50 ft How many feet would you save walking diagonally across the lot instead of walking length and width 50 feet 120feet 62 1202 502 62 14400 2500 c2 16900 J 16900 c 130ft Compared to walking 120 ft 50 ft 170 ft You would save walking 170 ft 7 130 ft 40 feet Example 5 A diagonal brace is to be placed in the wall of a room The height of the wall is 10 feet and the wall is 24 feet long See diagram below What is the length of the brace 10feet 24feet c2 102 242 c2 100576 c2 676 V02 l676 c 26feet Example 6 A television antenna is to be erected and held by guy wires If the guy wires are 40 ft from the base of the antenna and the antenna is 50 ft high what is the length of each guy wire 4M cl 402 502 02 1600 2500 c2 4100 02 l4100 3 c m 64feet Math History Excursion Commensurability and the Pythagorean Theorem The Pythagoreans as well as the Greeks believed that all distances and measurements were commensurable If two line segments a and b are commensurable then there exist a third segment that can be laid endtoend a whole number of times to produce segments that are equal in length to both line segment a and line segment b De nition If two segments are commensurable then there exists a third line segment c such that a mo and b no where n and m are integers For example let s use segments that measure 3 inches and 5 inches in length These segments are commensurable because there exist a third segment that measures 5 inches in length that can laid endtoend to 6 times to produce the 3 inch segment and 10 ten times to produce the 5 inch segment See illustration below a 5 inches b 3 inches c5 inches 0 segment c laid out 6 times is 3 inches o o o o o o o o o o c segmentc laid out 1 Otimes is 5 inches It is also true that two lengths are commensurable only if their quotient is a rational number A rational number is a number that can be expressed as the quotient of two integers A number that can not be expressed as a quotient of integers or a fraction is called an irrational number Also recall that the set of real numbers can be broken down into two separate or disj oints sets which are the rational numbers and the irrational numbers The d 39 I of the P Jquot Theorem 39 quot 39 J the Greek and Pythagorean notion that all measurements are commensurable For example suppose that we have square with side measuring 1 unit If we use the Pythagorean Theorem to nd the diagonal or hypotenuse we get that the hypotenuse is 5 See illustration below J5 J5 c21212 c2 ll c22 J7JE c This general observation produces two segments with lengths l and J5 that are not commensurable or incommensurable Notice if 1 andwE are commensurable then their quotient would be rational This would imply that g J5 is rational which is obviously false Therefore 5 or any other irrational number is going to produce a set of numbers that are incommensurable This fact alone was a contraction to Greek and Pythagorean concept that all where 39 39 However even when the Pythagoreans later discovered that the hypotenuse of a right triangle could equalJE they initially believed thanE was somehow rational In fact the Pythagoreans believed that mathematics and religion were one They also believe that all natural phenomena could be expressed by whole numbers or ratios of whole numbers For this reason they believed that xE could expressed as a ratio of whole numbers It is believed that it was Hippasus of Metapontum a Pythagorean who later discovered that number could be irrational while identifying the sides of the pentagram Later Theodorus of Cyrene proved that certain numbers where irrational but it wasn t until Exdoxus developed a theory of irrational ratios that a strong mathematical foundation for irrational numbers was created To nish up this section here are some examples of rational numbers and irrational numbers Examples of Rational Numbers 23 20 253333l 5 3 Note 3 is rational because can be written as and 3333 is equal to g Also not that any decimal number that repeats or terminates is rational Examples of Irrational Numbers EnH J3 7r e Note The value of pi 7239 and the Euler number 6 are irrational Example 6 Describe each number in the list as irrational or rational 2 471 3 Solution Irrational numbers 7239 Rational numbers 1

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