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Pythagoras' Theorem

Years ago, a man named Pythagoras found an amazing fact about triangles: If the triangle had a right angle (90°) ... ... and 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!

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 = 52 Calculating this becomes: 9 + 16 = 25 It 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:

a2 + b2 = c2

Now you can use algebra to find the missing values

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

a2 + b2 = c2 12 + 12 = c2 1 + 1 = c2 2 = c2 c2 = 2 c = √2 = 1.4142...

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)2 + (√5)2 = (√8)2 ? Does 3 + 5 = 8 ? Yes, it does! So this is a right-angled triangle

Any help/complains/enquiries please e-mail to ks3mathematics@gmail.com 

Hope you enjoyed :)

Circle Theorems

Angle in a Semicircle

An angle inscribed in a semicircle is always a right angle:

Why? Because: The inscibed angle 90° is half of the central angle 180° (Using "Angle at the Center Theorem" above)

Another Good Reason Why It Works We could also rotate the shape around 180° to make a rectangle! It is a rectangle, because all sides are parallel, and both diagonals are equal. And so its internal angles are all right angles (90°).

Tangent Angle A tangent is a line that just touches a circle at one point. It always forms a right angle with the circle's radius, as show