Pythagoras theorum
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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

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Pythagoras theorum

Pythagoras Theorum

Math 314


Pythagorean triples

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 c² = a² + b² Verify your answer given the #5 must be the largest value or c²

  • 5²= 3² + 4²

  • 25 = 9 + 16

  • 25=25 True 3,4,5 are Pythagorean triples


Label the triangle

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

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

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

Pythagorean Triples Fractions

  • 13 + 11

    143

  • 24(DO NOT REDUCE EVEN IF YOU CAN)

    143

  • Pythagorean triple is 24, 143 and 145

  • Pythagorean triple is numerator, denominator and denominator + 2.

  • Prove it or verify it.


Verify

Verify

  • Is 24, 143 and 145 Pythagorean triples?

  • c² = a² + b²

  • 145² = 24² + 143²

  • 21025 = 576 + 20449

  • 21025 = 21025 It works!


Example 2

Example #2

  • 2 and 4

  • 1 + 1

    2 4

  • 4 + 2

    8

  • 6

    8

  • Pythagorean triple is…

  • (6, 8, 10)


Even odd method faster

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

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

Another Method – Equation Method

  • Pick two natural numbers A + B such that A > B

  • A and B must be positive

  • 1) a² - b²

  • 2) 2ab

  • 3) a² + b²


Equation method to calculate pythagorean triple

Equation Method to Calculate Pythagorean Triple


Examples formula method

Examples – Formula Method

  • Generate a Pythagorean triple using the formula method

    A = 6; B = 1

  • Remember A²-B² 2AB A²+B²

    A² - B² = 36-1 = 35

  • 2AB = 2 (6) (1) = 12

  • A²+B² = 6² + 1² = 37

  • The numbers are (12, 35, 37)


More examples

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

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

Algebra and Pythagoras

  • How would you express the relationship between measures of the sides of the following right triangle

    5r

    3p

    4q

25r²= 9p² + 16 q² R = ?

R = 9p² + 16 q²

25


Calculating area of an isosceles triangle

Calculating Area of an Isosceles Triangle

1212

Cut triangle in half to calculate height

c² = a² + b²

  • 12² = 5² + a² (half of 10)

  • 144 = 25 + a²

  • 119 = a²10

  • a= 10.91

  • Area of isosceles triangle = base x height / 2

  • 10 x 10.91 / 2 = 54.55


Finding x with two missing variables

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

Calculating Height

  • We have two right angle triangles but we cannot get to the one with x directly so we need a middle step

  • 1st 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

Finding Height or k

x 9

k

7 5

81 = k² + 25

56 = k²

k = 7.48


Finding x

Finding x

x 9

7.48

7 5

x² = 7.48² + 7²

x² = 104.95

X = 10.24


Practice word problems

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

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 problem1

Word Problem

chair lift cable gondola cable

400

500

c² = 400² + 500² (find out c, then double to get g)

c² = 410000

C = 640 .31


Solution

Solution

  • Gondonla or G = 2c

  • G = 2 (640.31)

  • G = 1280.62

  • 1280.62 / 5 = 256.12 seconds


Word problems ladder

Word Problems - Ladder

  • 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.

  • Find the length of the ladder to the nearest tenth.

  • How many decimal places is tenth? hundredth, thousandth?


Diagram of ladder

Diagram of Ladder

  • 3m

8.4m

3.5m


Ladder solution

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

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

  • Irrational numbers have non – terminating, non repeating decimals. After the decimal, no pattern of numbers will repeat. Examples are…

  • Pie & square root of 2.


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