Pythagoras Pythagoras TheorumTheorum

Math 314Math 314

Pythagorean TriplesPythagorean Triples

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

The easiest is (3,4,5). Is this true?The easiest is (3,4,5). Is this true? If cIf c²² = a = a²² + b + b² ² Verify your answer given the #5 Verify your answer given the #5

must be the largest value or cmust be the largest value or c²² 55²= ²= 33² + ² + 44²² 25 = 9 + 1625 = 9 + 16 25=25 True 3,4,5 are Pythagorean triples25=25 True 3,4,5 are Pythagorean triples

Label the TriangleLabel the Triangle

Which of these numbers (3,4,5) Which of these numbers (3,4,5) must bemust be

the hypotenuse?the hypotenuse?

55

3 3

44 Does the placement of the 3, 4 or 5 make Does the placement of the 3, 4 or 5 make

a difference? a difference?

Creating other Creating other Pythagurus Triples. Your Pythagurus Triples. Your turn!turn!

Create 3 on your own Create 3 on your own and ask a friend to and ask a friend to guess what the other guess what the other one is? one is?

Label two out of the Label two out of the three legs and / or three legs and / or triangle. triangle.

Explain to them. Explain to them. Make it a decimal Make it a decimal (always two places)(always two places)

Pythagorean Triples with Pythagorean Triples with Fractions – Consecutive Fractions – Consecutive

Fraction MethodFraction Method Consider 11 and 13Consider 11 and 13 11 and 13 are consecutive odd numbers11 and 13 are consecutive odd numbers 11 + + 11

11 1311 13 Multiply denominators by each other (11 Multiply denominators by each other (11

* 13)* 13) Answer is 143. Therefore… Answer is 143. Therefore…

Pythagorean Triples Pythagorean Triples FractionsFractions

13 + 1113 + 11 143143

2424 (DO NOT REDUCE EVEN IF YOU CAN)(DO NOT REDUCE EVEN IF YOU CAN) 143143

Pythagorean triple is 24, 143 and 145Pythagorean triple is 24, 143 and 145 Pythagorean triple is numerator, Pythagorean triple is numerator,

denominator and denominator + 2.denominator and denominator + 2. Prove it or verify it. Prove it or verify it.

VerifyVerify

Is 24, 143 and 145 Pythagorean triples?Is 24, 143 and 145 Pythagorean triples? cc²² = a = a²² + b + b²² 145145²² = 24 = 24²² + 143 + 143²² 21025 = 576 + 2044921025 = 576 + 20449 21025 = 21025 It works!21025 = 21025 It works!

Example #2Example #2

2 and 4 2 and 4 11 + + 11 2 42 4 4 + 24 + 2 88 66 88 Pythagorean triple is… Pythagorean triple is… (6, 8, 10) (6, 8, 10)

Even Odd Method Even Odd Method (Faster)(Faster)

You get 2 consecutive even or odd You get 2 consecutive even or odd numbers; for example 7 & 9numbers; for example 7 & 9

Add them (7 + 9) = 16Add them (7 + 9) = 16 Multiply them (7 * 9) = 63Multiply them (7 * 9) = 63 Multiply them add 2 = 7 * 9 + 2 = 65Multiply them add 2 = 7 * 9 + 2 = 65 Triple is 16, 63, 65Triple is 16, 63, 65

Other ExamplesOther Examples

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

4, 6 4, 6 Answer: (10,24,26)Answer: (10,24,26) 8,108,10 Answer (18,80,82)Answer (18,80,82) 11,1311,13 Answer (24,143,145)Answer (24,143,145)

Another Method – Another Method – Equation MethodEquation Method

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

A and B must be positiveA and B must be positive

1) a1) a²² - b - b²² 2) 2ab2) 2ab 3) a3) a²² + b + b²²

Equation Method to Equation Method to Calculate Pythagorean Calculate Pythagorean TripleTriple

A = 11; B = 3

a² - b² a² - b² 11² - ² - 3²²

112

2ab2ab2 (11)(3)2 (11)(3)

6666

a² + b²a² + b²121 + 9121 + 9

130130

Examples – Formula Examples – Formula MethodMethod

Generate a Pythagorean triple using the Generate a Pythagorean triple using the formula methodformula method

A = 6; B = 1A = 6; B = 1 Remember ARemember A²²-B² 2AB A²+B²-B² 2AB A²+B²AA² - B² = 36-1 = 35² - B² = 36-1 = 35 2AB = 2 (6) (1) = 122AB = 2 (6) (1) = 12 A²+B² = 6² + 1² = 37A²+B² = 6² + 1² = 37 The numbers are (12, 35, 37)The numbers are (12, 35, 37)

More ExamplesMore Examples

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

DefinitionsDefinitions

Equilateral Triangle: All sides are equalEquilateral Triangle: All sides are equal Isosceles Triangle: Two sides are equalIsosceles Triangle: Two sides are equal Scalene: All sides are differentScalene: All sides are different What will you do when asked to calculateWhat will you do when asked to calculate Perimeter of Triangle?Perimeter of Triangle? Add up all the sidesAdd up all the sides Area of Triangle?Area of Triangle? Base x Height / 2Base x Height / 2

Algebra and Pythagoras Algebra and Pythagoras

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

5r5r

3p3p

4q4q

25r²= ²= 9p² + 16 q² ² + 16 q² R = ? R = ?

R = R = 9p² + 16 q²² + 16 q²

25

Calculating Area of an Calculating Area of an Isosceles TriangleIsosceles Triangle

1212 1212

Cut triangle in half to calculate heightCut triangle in half to calculate heightcc²² = a = a²² + b + b²² 12² = 5² + a² (half of 10) 12² = 5² + a² (half of 10) 144 = 25 + a²144 = 25 + a² 119 = a²119 = a² 1010 a= 10.91a= 10.91 Area of isosceles triangle = base x height / 2Area of isosceles triangle = base x height / 2 10 x 10.91 / 2 = 54.5510 x 10.91 / 2 = 54.55

Finding x with two Finding x with two missing variablesmissing variables

Triangle has different lengthsTriangle has different lengths

x 9x 9

7 57 5

Before calculating the x, find heightBefore calculating the x, find height

Therefore, do 2 Pythagoras's – double the fun!Therefore, do 2 Pythagoras's – double the fun!

Calculating HeightCalculating Height

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

11stst step is to find out missing value of x… step is to find out missing value of x… to figure that out use Pythagoras to figure that out use Pythagoras

x² = height² + 7²x² = height² + 7² You also know that 9² = height² and 5²You also know that 9² = height² and 5²

Finding Height or kFinding Height or k

x 9x 9

kk

7 57 5

81 = k² + 2581 = k² + 2556 = k²56 = k²k = 7.48k = 7.48

Finding xFinding x

x 9x 9

7.487.48

7 57 5x² = 7.48² + 7²x² = 7.48² + 7²

x² = 104.95x² = 104.95

X = 10.24X = 10.24

Practice – Word ProblemsPractice – Word Problems

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

Word ProblemWord Problem

chair lift cable gondola cablechair lift cable gondola cable

400 400

500500

If the gondola travels at 5m per second, how long If the gondola travels at 5m per second, how long with the gondola ride take?with the gondola ride take?

Word ProblemWord Problem

chair lift cable gondola cablechair lift cable gondola cable

400 400

500500

cc² = 400² + 500² (find out c, then double to get g)² = 400² + 500² (find out c, then double to get g)

c² = 410000c² = 410000

C = 640 .31 C = 640 .31

SolutionSolution

Gondonla or G = 2c Gondonla or G = 2c G = 2 (640.31)G = 2 (640.31) G = 1280.62G = 1280.62 1280.62 / 5 = 256.12 seconds1280.62 / 5 = 256.12 seconds

Word Problems - LadderWord Problems - Ladder

A ladder is leaning against a wall 8.4m A ladder is leaning against a wall 8.4m above the ground and extends 3m past above the ground and extends 3m past the top of the wall. The foot of the ladder the top of the wall. The foot of the ladder is 3.5m from the wall. is 3.5m from the wall.

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

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

Diagram of LadderDiagram of Ladder

3m3m

3.5m

8.4m

Ladder SolutionLadder Solution

c² = a² + b²c² = a² + b² c² = 8.4² + 3.5²c² = 8.4² + 3.5² c² = 70.56 + 12.25c² = 70.56 + 12.25 c² = 82.81c² = 82.81 C = 9.1C = 9.1 What do you do now?What do you do now? 9.1 + 3 = 12.1m is the length of the 9.1 + 3 = 12.1m is the length of the

ladder. ladder.

Rational NumbersRational Numbers

All rational numbers can be written in the form All rational numbers can be written in the form of fractions. For example; of fractions. For example;

14 = 14/114 = 14/1 0.72 = 72/1000.72 = 72/100 1.76 = 176/1001.76 = 176/100 These numbers have a zero or a group of These numbers have a zero or a group of

digits that repeat indefinitely. i.e. digits that repeat indefinitely. i.e. 1) 141) 14 2) 17.626262 or 17.622) 17.626262 or 17.62 3) 3.6666 or 3.63) 3.6666 or 3.6

Irrational NumbersIrrational Numbers

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

Pie & square root of 2. Pie & square root of 2.