The Pythagorean Theorem Helps Calculate The Size Of Which Shape?

Pythagoras' Theorem

Pythagoras

Over 2000 years ago there was an amazing discovery well-nigh triangles:

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

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

geometry/images/pyth1.js

... so the biggest square has the exact same area equally the other two squares put together!

Pythagoras

It is called "Pythagoras' Theorem" and can exist written in one brusque equation:

a² + b² = c²

pythagoras squares a^2 + b^2 = c^2

Annotation:

c is the longest side of the triangle
a and b are the other 2 sides

Definition

The longest side of the triangle is chosen the "hypotenuse", so the formal definition is:

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

Sure ... ?

Allow'south come across if it actually works using an instance.

Instance: A "3, four, 5" triangle has a correct angle in information technology.

Let's cheque if the areas are the same:

three² + 4^two = 5ⁱⁱ

Calculating this becomes:

9 + sixteen = 25

It works ... like Magic!

triangle 3 4 5 lego

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. (Merely call up information technology merely works on right angled triangles!)

How Exercise I Use it?

Write it down as an equation:

a² + bⁱⁱ = c^two

And so we use algebra to detect any missing value, every bit in these examples:

Example: Solve this triangle

right angled triangle 5 12 c

Start with: a^two + b² = cⁱⁱ

Put in what nosotros know: 5² + 12² = c^two

Calculate squares: 25 + 144 = c²

25+144=169: 169 = c²

Swap sides: c² = 169

Square root of both sides: c = √169

Calculate: c = 13

Multiples of 3,4,5

Read Builder'south Mathematics to run into practical uses for this.

Besides read well-nigh Squares and Square Roots to find out why √169 = 13

Example: Solve this triangle.

right angled triangle 9 b 15

Offset with: a^two + bⁱⁱ = cⁱⁱ

Put in what we know: nine^two + b² = xvⁱⁱ

Calculate squares: 81 + bⁱⁱ = 225

Have 81 from both sides: 81 − 81 + b² = 225 − 81

Summate: b² = 144

Square root of both sides: b = √144

Calculate: b = 12

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

Unit Square Diagonal

Start with: a^two + b² = c²

Put in what nosotros know: one^two + one² = c²

Calculate squares: one + 1 = c²

1+one=ii: 2 = c²

Swap sides: c² = two

Foursquare root of both sides: c = √2

Which is about: c = 1.4142...

It works the other way around, as well: when the 3 sides of a triangle make a² + bⁱⁱ = c^two , then the triangle is correct angled.

Case: Does this triangle have a Right Angle?

10 24 26 triangle

Does a² + b² = c² ?

a² + bⁱⁱ = x² + 24² = 100 + 576 = 676
c² = 26² = 676

They are equal, so ...

Yes, it does take a Right Angle!

Instance: Does an viii, 15, 16 triangle have a Correct Angle?

Does 8 ² + 15 ⁱⁱ = 16 ²?

8ⁱⁱ + xv² = 64 + 225 = 289,
but 16²= 256

So, NO, information technology does not have a Right Angle

Case: Does this triangle have a Right Bending?

Triangle with roots

Does a² + b² = c² ?

Does (√3)^two + (√5)² = (√8)^two ?

Does 3 + 5 = 8 ?

Yes, it does!

Then this is a right-angled triangle

And You Tin Prove The Theorem Yourself !

Get paper pen and scissors, then using the following blitheness every bit a guide:

Draw a right angled triangle on the newspaper, leaving enough of space.
Draw a square along the hypotenuse (the longest side)
Draw the aforementioned sized foursquare on the other side of the hypotenuse
Draw lines equally shown on the animation, like this:
Cut out the shapes
Conform them so that you tin can bear witness that the big square has the aforementioned expanse equally the two squares on the other sides

Another, Amazingly Unproblematic, Proof

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

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

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

The purple triangle is the important one.

Nosotros also have a proof by adding up the areas.

Historical Annotation: while we call it Pythagoras' Theorem, it was too known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived.

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