Years ago, a man named Pythagoras found an amazing fact about right triangles:

… If you made a square on each of the three sides, then …

… the biggest square had the exact same area as the other two squares put together!

This fact is called “**Pythagoras’ Theorem**” and can be written in one short equation:

a^{2} + b^{2} = c^{2}

Note:** c** is the **longest side** of the triangle and called “**hypotenuse**“

So in a right angled triangle:

the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let’s see if it really works using some examples.

**Example: Solve this triangle.**

**a**^{2} + b^{2} = c^{2}

^{2}+ b

^{2}= c

^{2}

5^{2} + 12^{2} = c^{2}

25 + 144 = c^{2}

169 = c^{2}

c^{2} = 169

c = √169

**c = 13 **

**Example: Solve this triangle.**

**a**^{2} + b^{2} = c^{2}

^{2}+ b

^{2}= c

^{2}

9^{2} + b^{2} = 15^{2}

81 + b^{2} = 225

Take 81 from both sides:

b^{2} = 144

b = √144

**b = 12 **

**Example: Does this triangle have a Right Angle?**

- 10
^{2}+ 24^{2}= 26^{2} - 100 + 576 =
**676** - 676 =
**676**

They are **equal**, so … Yes, it does have a Right Angle!

**Pythagorean Triples**

A “Pythagorean Triple” is a set of positive integers, **a**, **b** and **c** that fits the rule:

a^{2} + b^{2} = c^{2}

**Example: Does an 8, 15, 16 triangle have a Right Angle?**

Does 8^{2} + 15^{2} = 16^{2 }?

- 8
^{2}+ 15^{2}= 16^{2} - 64 + 225 =
**256**, ^{289 }=**256**

They are **NOT EQUAL**, so … it does not have a Right Angle

**Which one of the following is NOT a Pythagorean triple?**

**A:**7, 24, 25

**B:**8, 15, 17

**C:**9, 12, 15

**D:**10, 16, 19

Pythagorean Triples are sets of whole numbers which fit the rule: ** a ^{2} + b^{2} = c^{2}**

In A, 7^{2} + 24^{2} = 49 + 576 = 625 = 25^{2} ⇒ 7, 24, 25 **is** a Pythagorean triple.

In B, 8^{2} + 15^{2} = 64 + 225 = 289 = 17^{2} ⇒ 8, 15, 17 **is** a Pythagorean triple.

In C, 9^{2} + 12^{2} = 81 + 144 = 225 = 15^{2} ⇒ 9, 12, 15 **is** a Pythagorean triple.

In D, 10^{2} + 16^{2} = 100 + 256 = 356 ≠ 19^{2} ⇒ 10, 16, 19 **is not** a Pythagorean triple.

**If (x, 40, 41) is a Pythagorean triple, what is the value of x?**

Pythagorean Triples are sets of positive integers which fit the rule: **a ^{2} + b^{2} = c^{2}**

Replace **a** by **x**, **b** by **40** and **c** by **41**

⇒ x^{2} + 40^{2} = 41^{2}

⇒ x^{2} + 1,600 = 1,681

⇒ x^{2} = 1,681 – 1,600

⇒ x^{2} = 81

⇒ x = √81 = 9

**What follows is a PowerPoint presentation that will help you understand the content of the lesson in a better and easier way**, (*pps*).

Hi Mr,

how are you ?

thank you for this flash game i solved it and got all my answers correct

Hi Abdulazeez

I’m so happy to see you working like that

Keep up the good work

thank you mister ….. this will really help for tomorrow’s quiz

My pleasure :), wish the best in your quiz

thank you Mr.Elhaj (:

you’r welcome, Glad i could help

thank u MR.

thank you very much Mr.ahmed this was very helpful to understand the lesson

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mr.ahmed thank u very very very much for this website some people can learn from this and some people could learn in a book really some people improved from it thank u again

you’r very very welcome dear Basel, Glad i could help

Thanks you so much sir, I couldn’t have understood this lesson properly for the quiz without the website, And you of course ,I appreciate the hard work you’re having just to help us.

_ Ibrahim Ali