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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

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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 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

• 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)

### 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

• Is 24, 143 and 145 Pythagorean triples?

• c² = a² + b²

• 145² = 24² + 143²

• 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

• 8,10

• 11,13

### 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

### 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

• 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

25r²= 9p² + 16 q² R = ?

R = 9p² + 16 q²

25

### 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

• 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

• 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

x 9

k

7 5

81 = k² + 25

56 = k²

k = 7.48

x 9

7.48

7 5

x² = 7.48² + 7²

x² = 104.95

X = 10.24

### 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

c² = 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

• 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?

• 3m

8.4m

3.5m

• 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.