Pythagoras Theorum

Math 314

Pythagorean Triples

      Can you think of 3 natural numbers that would work in a right angled triangle?

The easiest is (3,4,5). Is this true?

If c ² = a ² + b ² Verify your answer given the #5 must be the largest value or c ² 5 ²= 3 ² + 4 ² 25 = 9 + 16 25=25 True 3,4,5 are Pythagorean triples

Label the Triangle

 Which of these numbers (3,4,5) must be the hypotenuse?

5 3 4  Does the placement of the 3, 4 or 5 make a difference?

Creating other Pythagurus Triples. Your turn!

   Create 3 on your own and ask a friend to guess what the other one is? Label two out of the three legs and / or triangle. Explain to them. Make it a decimal (always two places)

Pythagorean Triples with Fractions – Consecutive Fraction Method

   Consider 11 and 13 11 and 13 are consecutive odd numbers 1 + 1 11 13   Multiply denominators by each other (11 * 13) Answer is 143. Therefore…



Pythagorean Triples Fractions

13 + 11 143  24 143

(DO NOT REDUCE EVEN IF YOU CAN)

   Pythagorean triple is 24, 143 and 145

Pythagorean triple is numerator, denominator and denominator + 2.

Prove it or verify it.

Verify

     Is 24, 143 and 145 Pythagorean triples?

c ² = a ² + b ² 145 ² = 24 ² + 143 ² 21025 = 576 + 20449 21025 = 21025 It works!

Example #2

      2 and 4 1 + 1 2 4 4 + 2 8 6 8 Pythagorean triple is… (6, 8, 10)

Even Odd Method (Faster)

     You get 2 consecutive even or odd numbers; for example 7 & 9 Add them (7 + 9) = 16 Multiply them (7 * 9) = 63 Multiply them add 2 = 7 * 9 + 2 = 65 Triple is 16, 63, 65

Other Examples

       Generate a Pythagorean triple using the even – odd seed method.

4, 6 Answer: (10,24,26) 8,10 Answer (18,80,82) 11,13 Answer (24,143,145)

Another Method – Equation Method

 Pick two natural numbers A + B such that A > B  A and B must be positive    1) a ² - b ² 2) 2ab 3) a ² + b ²

Equation Method to Calculate Pythagorean Triple

A = 11; B = 3 a² - b² 11 ² - 3² 112 2ab 2 (11)(3) 66 a² + b² 121 + 9 130

Examples – Formula Method

 Generate a Pythagorean triple using the formula method A = 6; B = 1  Remember A ² B² 2AB A²+B² A ² - B² = 36-1 = 35  2AB = 2 (6) (1) = 12  A²+B² = 6² + 1² = 37  The numbers are (12, 35, 37)

More Examples

      A = 6 ; B = 2 Solution (24,32,40) A = 6 ; B = 3 Solution (27,36,45) A = 12 ; B = 1 Solution ( 24, 143, 145)

Definitions

        Equilateral Triangle: All sides are equal Isosceles Triangle: Two sides are equal Scalene: All sides are different What will you do when asked to calculate Perimeter of Triangle?

Add up all the sides Area of Triangle?

Base x Height / 2

Algebra and Pythagoras

 How would you express the relationship between measures of the sides of the following right triangle 5r 3p 4q

25r

²=

9p

² + 16 q² R = ? R =

9p

² + 16 q²

25

Calculating Area of an Isosceles Triangle

12 12 Cut triangle in half to calculate height c ²       = a ² + b ² 12² = 5² + a² (half of 10) 144 = 25 + a² 119 = a² 10 a= 10.91

Area of isosceles triangle = base x height / 2 10 x 10.91 / 2 = 54.55

Finding x with two missing variables

 Triangle has different lengths x 9 7 5 Before calculating the x, find height Therefore, do 2 Pythagoras's – double the fun!

Calculating Height

    We have two right angle triangles but we cannot get to the one with x directly so we need a middle step 1 st step is to find out missing value of x… to figure that out use Pythagoras x² = height² + 7² You also know that 9² = height² and 5²

Finding Height or k

x 9 k 7 5 81 = k² + 25 56 = k² k = 7.48

Finding x

x² = 7.48² + 7² x² = 104.95

X = 10.24

x 9 7.48

7 5

Practice – Word Problems

 Both a chair lift and a gondola are used to transport skiers to the top of a ski hill. The length of the gondola cable is twice the length of the chair lift cable. The situation is represented by

Word Problem

chair lift cable gondola cable 400 500 If the gondola travels at 5m per second, how long with the gondola ride take?

Word Problem

chair lift cable gondola cable 400 500 c ² = 400² + 500² (find out c, then double to get g) c² = 410000 C = 640 .31

Solution

    Gondonla or G = 2c G = 2 (640.31) G = 1280.62

1280.62 / 5 = 256.12 seconds

Word Problems - Ladder

  Find the length of the ladder to the nearest tenth.  A ladder is leaning against a wall 8.4m above the ground and extends 3m past the top of the wall. The foot of the ladder is 3.5m from the wall. How many decimal places is tenth? hundredth, thousandth?



Diagram of Ladder

3m

8.4m

3.5m

Ladder Solution

       c² = a² + b² c² = 8.4² + 3.5² c² = 70.56 + 12.25

c² = 82.81

C = 9.1

What do you do now?

9.1 + 3 = 12.1m is the length of the ladder.

Rational Numbers

        All rational numbers can be written in the form of fractions. For example; 14 = 14/1 0.72 = 72/100 1.76 = 176/100 These numbers have a zero or a group of digits that repeat indefinitely. i.e. 1) 14 2) 17.626262 or 17.62

3) 3.6666 or 3.6

Irrational Numbers

 Irrational numbers have non – terminating, non repeating decimals. After the decimal, no pattern of numbers will repeat. Examples are…  Pie & square root of 2.