Pythagoras Theorem Presentation

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Pythagoras

Over 2000 years ago there was an amazing discovery about triangles:

This is a PowerPoint math or maths presentation on The Theorem of Pythagoras. We start by looking at some of characters included in Raphael's fresco 'The School of Athens' from the Vatican Museum. We see Plato and Aristotle as the central characters and then show others including Socrates, Euclid and Pythagoras. Pythagorean Theorem 9 16 25 There are at least 80 different ways to prove the Pythagorean theorem Link to AgileMinds #13, Patty Paper Proof, 1-8 Pythagorean Theorem Find the missing measures √27 12.7 Converse of the Pythagorean Th. Determine whether the sides of these triangles could form a right triangle 9, 12, 15 4√3, 4, 8 5, 8, 9 yes yes. Introducing Pythagoras. Lesson presentations and activities. The activity book allows them to investigate pythagoras theorem for themselves and discover the link between a, b and c. Once started, pupils can move through the activities independently, although the powerpoint allows for more teacher-led episodes if needed.

When a triangle has a right angle (90°) ...

... and squares are made on each of the three sides, ...

... then the biggest square has the exact same area as the other two squares put together!

It is called 'Pythagoras' Theorem' and can be written in one short equation:

a2 + b2 = c2

Note:

• c is the longest side of the triangle
• a and b are the other two sides

Definition

The longest side of the triangle is called the 'hypotenuse', so the formal definition is:

In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.

Sure ... ?

Let's see if it really works using an example.

Example: A '3,4,5' triangle has a right angle in it.

 Let's check if the areas are the same:32 + 42 = 52Calculating this becomes: 9 + 16 = 25It works ... like Magic!

Why Is This Useful?

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!)

How Do I Use it?

Write it down as an equation:

Then we use algebra to find any missing value, as in these examples:

Example: Solve this triangle Put in what we know:52 + 122 = c2
25+144=169:169 = c2
Square root of both sides:c = √169

You can also read about Squares and Square Roots to find out why 169 = 13

Example: Solve this triangle.

Put in what we know:92 + b2 = 152
Take 81 from both sides: 81 − 81 + b2 = 225 − 81 Square root of both sides:b = √144

Example: What is the diagonal distance across a square of size 1?

Put in what we know:12 + 12 = c2
1+1=2: 2 = c2
Square root of both sides:c = √2

It works the other way around, too: when the three sides of a triangle make a2 + b2 = c2, then the triangle is right angled.

Example: Does this triangle have a Right Angle?

Does a2 + b2 = c2 ?

• a2 + b2 = 102 + 242 = 100 + 576 = 676
• c2 = 262 = 676

They are equal, so ...

Yes, it does have a Right Angle!

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

Does 82 + 152 = 162 ?

• 82 + 152 = 64 + 225 = 289,
• but 162 = 256

So, NO, it does not have a Right Angle

Example: Does this triangle have a Right Angle?

Does a2 + b2 = c2 ?

Does 3 + 5 = 8 ?

Yes, it does!

Pythagoras Theorem Presentation Example

So this is a right-angled triangle

And You Can Prove The Theorem Yourself !

Get paper pen and scissors, then using the following animation as a guide:

• Draw a right angled triangle on the paper, leaving plenty of space.
• Draw a square along the hypotenuse (the longest side)
• Draw the same sized square on the other side of the hypotenuse
• Draw lines as shown on the animation, like this:
• Cut out the shapes
• Arrange them so that you can prove that the big square has the same area as the two squares on the other sides

Another, Amazingly Simple, Proof

Here is one of the oldest proofs that the square on the long side has the same area as the other squares.

Watch the animation, and pay attention when the triangles start sliding around.

You may want to watch the animation a few times to understand what is happening.

The purple triangle is the important one.

 becomes

We also have a proof by adding up the areas.

 Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived !

Activity: Pythagoras' Theorem
Activity: A Walk in the Desert