Pythagoras Theorem

EYAD HAKAMI

MARCH 1, 2015

c

b

a 𝑎2+𝑏2=𝑐2

BackgroundPythagoras was a Greek Philosopher and Mathematician who lived in the fifth century B.C. He's mostly known for the Pythagorean Theorem, which is said to be the most famous theorem in Mathematics.

Fit a pole inside a truck trailer!

Suppose you have a truck with a trailer that is 5 meters long, 3 meters high and 2 meters in width. What is the longest pole that can fit inside this truck?

We will find out at the end of this lesson!

Terminology and Definitions

A right triangle is a triangle with a 90° angle. The hypotenuse is the side of the triangle opposite to that angle, and its legs are the other two sides.

hypoten

useleg

leg

Statement of the Theorem

The Pythagorean Theorem states the relationship between the sides of a right triangle. If a right triangle has legs with lengths a and b, and a hypotenuse with length c, then

c

b

a

Look at it differently!

Note here that the area of the smaller square is equal to the area of the bigger square minus the area of the four right triangles. The sides of the bigger square have length The sides of the smaller square are of length , which is also the hypotenuse of the triangles. Hence the area of the smaller square is . Simplifying this yields that

b

a

a

a

b

a b

c

ac

cc

ExamplesQuestion: In the graph shown, find the length of the hypotenuse c.

Solution: Using Pythagoras Theorem, So ft.

3ft

4ftc

ExamplesQuestion: Find b.

Solution: . This can be rewritten as

or

So the length of the third size is

10

8

b

Inverse of the Statement

The inverse of the statement is also true. That's to say that a triangle with sides a, b and c satisfying must be a right triangle.

5

Example: Verify that a triangle with sizes 5, 12 and 13 is a right triangle.

Solution: Note that Since it satisfies Pythagoras theorem, the triangle must be a right-triangle.

12

135

Back to the truck

To be able to find the longest pole to fit inside a truck trailer of dimensions 5, 3 and 2 meters, we need to apply Pythagoras Theorem. The solution is explained in the next slide.

Solution

It’s obvious that the red line represents the longest possible pole that can fit inside the trailer. Note that the red line is the hypotenuse of the triangle with the blue line as one leg and the height of the trailer as the other leg. So we first find the length of the blue line.

Now we have Hence the longest pole is meters .

x

y

5m2m

3m

Exercises1- A right triangle with leg of size 5 and 7 length units. What’s the length of its hypotenuse.

2- Find the diagonal of a square with side length of 9 meters.

3- A triangle with sides 7, 24 and 25. Is it a right-triangle. Justify your answer.