Pythagorean Theorem

Pythagorean Theorem

A right triangle has a 90^。angle. The longest side of the triangle, c is opposite to the right angle and it is called the hypotenuse.  The other two sides, a and b, are called legs.

Pythagorean Theorem:


For a right triangle ABC with sides a, b and c, and ∠C = 90^。

c² = a² + b²                                      

The Pythagorean Theorem is the most famous and most important result in geometry. It allows us to compute the unknown length in a right-angled triangle, if given the other two sides.

Example: Calculate the length of the unknown side.

c² = 3² + 4²

    = 9 + 16 = 25

c = √25 = 5 m

Example: Calculate the length of the unknown side.

10² = 8² + b²

100 = 64 + b²

100-64 = 64 + b²-64

36 = b²

B =√36 = 6 cm

Practice:
Calculate the length of the unknown side, to one decimal place.

          15² = 13² + r²






Pythagorean Theorem Extension

After introducing Pythagorean Theorem, I really want to talk about one convenient way to calculate the length of the three sides in a right angle triangle. 

If you have done a lot of calculations on Pythagorean Theorem, you would have noticed that there is a ratio for the three sides that is always true. 

The ratio is a : b : c = 3 : 4 : 5, with 5 being the c (hypotenuse) side.

For example:

The length of side a (adjacent) and b (opposite) are 6cm and 8cm. Find the hypotenuse. 

Then you can find out the hypotenuse by:

       a : b : c = 3 : 4: 5 

=>  a : b : c = 6 : 8 : 10

Since 3 : 4 = 3 * 2 : 4 * 2 = 6 : 8

Therefore 5 * 2 = 10 cm.

a : b : c = 6 : 8 : 10

Ans: Hypotenuse is 10 cm.

This way of calculation is NOT applicable to Isosceles Right Angle Triangle.