### Grade9: Lesson 5-7: The Pythagorean Theorem

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*).

November 23, 2011 at 8:03 pm

Hi Mr,

how are you ?

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

November 25, 2011 at 3:51 pm

Hi Abdulazeez

I’m so happy to see you working like that

Keep up the good work

January 30, 2012 at 4:39 pm

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

January 30, 2012 at 9:30 pm

My pleasure :), wish the best in your quiz

January 13, 2013 at 10:01 pm

thank you Mr.Elhaj (:

January 19, 2013 at 3:12 pm

you’r welcome, Glad i could help

January 20, 2013 at 5:40 pm

thank u MR.

January 20, 2013 at 5:54 pm

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

February 8, 2013 at 1:10 am

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March 8, 2013 at 1:32 pm

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

March 18, 2013 at 4:18 pm

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