Section 7.1 The Pythagorean Theorem. Right Triangles

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1 Setion 7. The Pythagorean Theorem It is better wither to be silent, or to say things of more value than silene. Sooner throw a pearl at hazard than an idle or useless word; and do not say a little in many words, but a great deal in a few. Pythagoras Right Triangles The origins of right triangle geometry an be traed bak to 3000 BC in Anient Egypt. The Egyptians used speial right triangles to survey land by measuring out 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 onept of a variable. The onept of a variable was developed by a mathematiian named Diophantes of Alexandria. The Egyptians most studied speifi examples of right triangles. For example, the Egyptians use ropes to measure out distanes to form right triangles that were in whole number ratios. In the next illustration, it is demonstrated how a right triangle an be form using ropes to reate a right angle. 5 knots 3 knots 4 knots Using ropes that had knots that where equality spaed, the Egyptians ould measure out right angles by making a right angle or other right triangles with the rope. It wasn t until around 500 BC, when a Greek mathematiian name Pythagoras disovered that there was a formula that desribed 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. B b A a C = a + b

2 Example Determine if the triangle measured out by ropes ontains a right angle. If you ount the number of knots on eah side of the triangle you get a ratio of knots 6 knots 8 knots Substituting these values into the Pythagorean Theorem using 0 as the hypotenuse and the other two sides as the legs, you an determine if the triangle is a right triangle. = a + b 0 = 6 0 = = Sine the formula heks, the triangle is a right triangle giving the triangle a right angle. Here are some examples of how the Pythagorean Theorem an be use to find the missing side of a right triangle.

3 Example Suppose the two legs of a right triangle are 5 units and units, find 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 =, and solve for = 5 + = = 69 = = 3 69 Example 3 Suppose that the hypotenuse of a right triangle is 6 units and one leg is 0 units, find the measure of the missing 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 = 0, = 6, find 6 = = 00 + b = b 576 = b + b b = 575 b = 4 b Appliations of the Pythagorean Theorem The Pythagorean Theorem is widely used in many fields. Many problems an be modeled or represented by right triangles, so the Pythagorean Theorem is used to solve many types of problems. When a problem is model by a right triangle, values an be assigned to the sides of the triangle and the unknown value an be found by solving for the missing side of the triangle. Here are some examples of appliations of right triangles and the Pythagorean Theorem

4 Example 4 An empty lot is 0 ft by 50 ft. How many feet would you save walking diagonally aross the lot instead of walking length and width? 50 feet 0 feet = = = 6900 = = 30 ft 6900 Compared to walking 0 ft + 50 ft = 70 ft You save walking 70 ft 30 ft = 40 feet Example 5 A diagonal brae is to be plaed in the wall of a room. The height of the wall is 0 feet and the wall is 4 feet long. (See diagram below) What is the length of the brae? 0 feet 4 feet = = = 676 = 676 = 6 feet

5 Example 6 A television antenna is to be ereted 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 eah guy wire? 50 ft = ft + 50 = = 400 = feet Math History Exursion: Commensurability and the Pythagorean Theorem The Pythagoreans as well as the Greeks believed that all distanes and measurements were ommensurable. If two line segments a and b are ommensurable, then there exist a third segment that an be laid end-to-end a whole number of times to produe segments that are equal in length to both line segment a and line segment b. Definition: If two segments a and b are ommensurable, then there exists a third line segment suh that a = m and b = n where n and m are integers. For example, let s use segments that measure 3 inhes and 5 inhes in length. These segments are ommensurable, beause there exist a third segment that measures.5 inhes in length that an laid end-to-end to 6 times to produe the 3 inh segment and 0 times to produe the 5 inh segment. (See illustration below)

6 a = 5 inhes b = 3 inhes =.5 inhes segment laid out 6 times is 3 inhes segment laid out 0 times is 5 inhes It is also true that two lengths are ommensurable only if their quotient is a rational number. A rational number is a number that an be expressed as the quotient of two integers. A number that an not be expressed as a quotient of integers or a fration is alled an irrational number. Also reall that the set of real numbers an be broken down into two separate or disjoints sets whih are the rational numbers and the irrational numbers. The development of the Pythagorean Theorem invalidated the Greek and Pythagorean notion that all measurements are ommensurable. For example, suppose that we have square with side measuring unit. If we use the Pythagorean Theorem to find the diagonal or hypotenuse, we get that the hypotenuse is. (See illustration below) = + = + = = =

7 This general observation produes two segments with lengths and that are not ommensurable or inommensurable. Notie, if and are ommensurable then their quotient would be rational. This would imply that = is rational whih is obviously false. Therefore, or any other irrational number is going to produe a set of numbers that are inommensurable. This fat alone was a ontration to Greek and Pythagorean onept that all measurements where ommensurable. However, even when the Pythagoreans later disovered that the hypotenuse of a right triangle ould equal, they initially believed that was somehow rational. In fat, the Pythagoreans believed that mathematis and religion were one. They also believe that all natural phenomena ould be expressed by whole numbers or ratios of whole numbers. For this reason, they believed that ould be expressed as a ratio of whole numbers. This assumption would later be proven to be false. Brief History of Irrational Numbers ) It is believed that it was Hippasus of Metapontum, a Pythagorean, who later disovered that number ould be irrational while identifying the sides of the pentagram. ) Theodorus of Cyrene proved that ertain numbers where irrational, but it wasn t until Exdoxus developed a theory of irrational ratios that a strong mathematial foundation for irrational numbers was reated. Examples of Rational Numbers,3, 5,0,.5, , 3 Note: 3 is rational beause an be written as 3 and is equal to 3 Also not that any deimal number that repeats or terminates is rational. Examples of Irrational Numbers,, 5, π,e Note: The value of pi π and the Euler number e are irrational. Now let s develop our understanding of rational numbers and irrational numbers by identifying number as irrational or rational from a list of numbers.

8 Example 6 Desribe eah number in the list as irrational or rational. 7,, π, 3 Solution: The number 7 is irrational beause it is radial expression that annot be simplified. Remember radial expressions that annot be simplified annot be expressed as a fration. The numberπ is irrational sine it is a number that does not terminate or repeat whih implies that it annot be expressed as a fration. The number is rational beause it an be expressed as a fration. The number 3 is obviously rational sine it is already in the fration form.

9 Exerises ) Find the missing side of right triangle ABC where a and b are the legs and is the hypotenuse. a) a = 0, b = 4, find b) a =, = 0, find b ) a = 8, b = 8, find d) a = 5, b =, find ) Suppose that the hypotenuse of a right triangle is 0 units and one leg is 8 units, find the measure of the other leg 3) Suppose the two legs of a right triangle are 5 units and 7 units respetively, find the length of the hypotenuse. 4) A retangular shaped lot is 80 ft by 60 ft. How many feet would you save walking diagonally aross the lot instead of walking length and width? 5) Find the length diagonal of a retangle that is 30 ft by 40 ft. 6) A diagonal brae is to be plaed in the wall of a room. The height of the wall is 8 feet and the wall is 0 feet long. (See diagram below) What is the length of the brae? 8 feet 0 feet 7) A television antenna is to be ereted and held by guy wires. If the guy wires are 40 ft from the base of the antenna and the antenna is 70 ft high, what is the length of the guy wire? h = 70 feet 40 feet 40 feet