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

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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:

a2 + b2 = c2

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.

### a2 + b2 = c2

52 + 122 = c2

25 + 144 = c2

169 = c2

c2 = 169

c = √169

c = 13

Example: Solve this triangle.

### a2 + b2 = c2

92 + b2 = 152

81 + b2 = 225

Take 81 from both sides:

b2 = 144

b = √144

b = 12

Example: Does this triangle have a Right Angle?
Does a2 + b2 = c2 ?
• 102 + 242 = 262
• 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:

a2 + b2 = c2

Examples of Pythagorean Triples are:

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

Does 82 + 152 = 162 ?

• 82 + 152 = 162
• 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:  a2 + b2 = c2

In A, 72 + 242 = 49 + 576 = 625 = 252 ⇒ 7, 24, 25 is a Pythagorean triple.
In B, 82 + 152 = 64 + 225 = 289 = 172 ⇒ 8, 15, 17 is a Pythagorean triple.
In C, 92 + 122 = 81 + 144 = 225 = 152 ⇒ 9, 12, 15 is a Pythagorean triple.
In D, 102 + 162 = 100 + 256 = 356 ≠ 192 ⇒ 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: a2 + b2 = c2

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

⇒ x2 + 402 = 412

⇒ x2 + 1,600 = 1,681

⇒ x2 = 1,681 – 1,600

⇒ x2 = 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).

## 2 thoughts on “Grade9: Lesson 5-7: The Pythagorean Theorem”

omar mahdi said:
January 30, 2012 at 4:39 pm

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

noureldin rashed said:
January 20, 2013 at 5:54 pm

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