1. Pythagoras Theorum
Math 314

2. 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 =
25=25 True 3,4,5 are Pythagorean triples

3. 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?

4. 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)

5. 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…

6. Pythagorean Triples Fractions
143
24 (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.

7. Verify Is 24, 143 and 145 Pythagorean triples? c² = a² + b²
145² = 24² + 143²
21025 =
21025 = It works!

8. Example #2 2 and 4 1 + 1 2 4 4 + 2 8 6 Pythagorean triple is…
4 + 2 8 6
Pythagorean triple is…
(6, 8, 10)

9. 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 * = 65
Triple is 16, 63, 65

10. 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)

11. 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

12. Equation Method to Calculate Pythagorean Triple

13. 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)

14. More Examples A = 6 ; B = 2 Solution (24,32,40) A = 6 ; B = 3

15. 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

16. 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

17. Calculating Area of an Isosceles Triangle
Cut triangle in half to calculate height
c² = a² + b²
12² = 5² + a² (half of 10)
144 = 25 + a²
119 = a²
a= 10.91
Area of isosceles triangle = base x height / 2
10 x / 2 = 54.55

18. Finding x with two missing variables
Triangle has different lengths x
Before calculating the x, find height
Therefore, do 2 Pythagoras's – double the fun!

19. 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

20. Finding Height or k x k
81 = k² + 25
56 = k²
k = 7.48

21. Finding x x
7.48
x² = 7.48² + 7²
x² =
X = 10.24

22. 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

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

24. Word Problem chair lift cable gondola cable 400 500
c² = 400² ² (find out c, then double to get g)
c² =
C =

25. Solution Gondonla or G = 2c G = 2 (640.31) G = 1280.62
/ 5 = seconds

26. 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?

27. Diagram of Ladder 3m
8.4m
Emphasize the ladder part
3.5m

28. Ladder Solution c² = a² + b² c² = 8.4² + 3.5² c² = 70.56 + 12.25
What do you do now?
= 12.1m is the length of the ladder.
Next slide represent 14,

29. 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) or 17.62
3) or 3.6

30. 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.