 # Looking For Pythagoras

 Special right triangles: A triangle with angles 30, 60 and 90 degrees will have side lengths that satisfy the Pythagorean relationship; a triangle with angles 45, 45 and 90 degrees will have side lengths that satisfy the Pythagorean relationship. The side lengths of any 30-60-90 triangle are in the ratio 1:√3:2; the side lengths of any 45-45-90 triangle are in the ratio 1: 1: √2. 9. To see why a 30-60-90 triangle has sides in a particular ratio we first examine a 60-60-60 triangle with each side length 2 units. Notice that the altitude (at right angles to the base) bisects the base into two lengths , each 1 unit, creating two 30-60-90 triangles. The sides of this 30-60-90 triangle satisfy the Pythagorean relationship and so, 12 + x2 = 22, so x2 = 3, so x = √3. The side lengths are 1, √3, 2 units 10. Triangle ABC, sketched below, has angles 30, 60, 90 degrees, and the shortest side length is 3 units. What are the other side lengths? This triangle is a scaled up similar copy of the triangle in example 9. (See Stretching and Shrinking.) The scale factor is 3. So, the lengths are 3(1), 3(√3), and 3(2) units, or 3, 3√3, 6 units. 11. Triangle PQR is a 45-45-90 triangle. The hypotenuse is 5 units long. How long are the other sides? The ratio of side lengths for a 45-45-90 triangle is 1:1:√2. In this triangle, which is a similar copy of every other 45-45-90 triangle, the ratio is p: r: 5, where p and r are the equal sides. We can think of using a scale factor of 5/(√2) to scale up a triangle with sides that measure 1, 1, √2 units. This will create a triangle with sides that measure 5/(√2), 5/(√2), 5 units, or approximately 3.5, 3.5, 5 (Note: students could also find the sides by using some algebraic reasoning . P2 +r2 = q2, so p2 + p2 = 25, so 2p2 = 25, so p2 = 12.5, so p = √12.5. or approximately 3.5 units.) q = 5 Rational numbers: are any numbers that can be written in the form a/b where a and b are integers, but b can not be zero . Students can think of these as anything that can be written as a positive or negative fraction.Note: every rational number can be written as a decimal, either terminating or repeating . (See Vocabulary, Bits and Pieces III.)   Irrational numbers: are numbers that can NOT be written in the form a/b where a and b are integers. Non-repeating, non-terminating decimals, and square roots that do not work out exactly and ⇐ are examples of irrational numbers. Note: since numbers like √2 and √5 are irrational any decimal approximation will be inexact, no matter how many decimal places we use. √2 = 1.4142… and √5 = 2.2360… The decimal approximations never terminate and never repeat. If they did terminate or repeat then these decimals could be written as rational numbers; but √2 and √5 are irrational numbers. Using the format √2 is exact, whereas 1.4142 is a very accurate, but inexact, approximation.   Real #’s: are all the numbers which are either rational or irrational. Note: every number that students know about at this stage is a real number . In High School they will meet other kinds of numbers, such as complex numbers . 12. Which of these numbers are rational numbers: 2, 2.4, 0.1111…, -9, 2 1/3, 17/5, -2/7?ALL of these numbers are rational. They CAN all be written as a/b. 2 = 2/1 2.4 = 24/10 (every terminating decimal can be written as a fraction with a power of 10 for a denominator ) 0.1111… = 1/9 (see below) -9 = -9/1 2 1/3 = 7/3 17/5 is already in the “a/b” format. -2/7 is already in the “a/b” format. From the above examples we can conclude that any integer, any positive or negative fraction , or mixed number, and any terminating decimal can be written as a rational number. 13. a. Write 1/9 as a decimal. Every fraction can be thought of as a division . So 1/9 can be thought of as 1 ÷ 9. We can set this up as a division, 1.0000 ÷ 9, and get the decimal answer, 0.1111… (See Bits and Pieces III for decimal division.) b. Write 0.121212…as a rational number. We can think of this as an algebra problem. X = 0.121212… So, 100x = 12.121212… So, 100x – x = 12.121212…. – 0.121212…. = 12. (Notice there is no repeating part now.) So, 99x = 12. So, x = 12/99. This strategy could have been used for any repeating decimal. Any repeating decimal can be written as a rational number. 14. Give an example of a non-terminating and non-repeating decimal. 0.3 is a terminating decimal. 0.333…is a repeating decimal. But 0.32332333233332… has a pattern which neither terminates nor repeats. Thus 0.32332333233332…is an irrational number.
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