pythagorean theorem 2 algebraic proofs. pythagoras’ proof

Pythagorean Theorem 2 Algebraic Proofs

Pythagoras’ Proof

What figures do you see in this diagram? Whatrelationship do you see between the figures?

The figure contains 2 squares and 4 right triangles. The hypotenuse of each right triangle are the sides of the small square. Each side of the large square is the sum of a short leg and long leg of a right triangle. The area of the big square is the sum of the area of the small square and the 4 right triangles.

Write a verbal equation using the area relationships

shown in the figure.

Area of Large Square = Area of Small Square + Four times

Area of Triangle.

Write an expression for the area of the large square.

Area of Large Square

Write an expression for the area of the small square.

Area of Small Square

Write an expression for the area each right triangle.

Area of Right Triangle

We have already compared the areas as:Area of Big Square = Area of Small Square

+ 4 Right Triangles

Now fill in the expressions we have written for each figure.

Now simplify.

Simplify.

How does this equation apply to the right triangles in the figure?

Bhaskara’s First Proof

What figures do you see and what relationships do you see in the diagram?

Assume right angles.

A large square, a small square, and 4 right triangles. The sides of the big square are the hypotenuses of the right triangles. Each side of the small square is the difference of the legs of the right triangle. (*explored in next slides)The area of the large square is the sum of the small square and 4 right triangles.

ab

Let’s call the longer leg of each right triangle, a, and the shorter leg, b.

ab

Let’s also label b in the left side triangle.Now we can see that the side of the smallersquare is a – b.

b

Write a verbal equation showing the relationshipbetween the areas of the figures shown.

Area of Large Square = Area of Small Square + Area of 4 Right Triangles.

Write an expression for the area of the large square.

Area of Large Square

Write an expression for the area of each right triangle.

Area of Right Triangle

Write an expression for the area of the small square.

Area of Small Square

Now fill in the expressions:Area of Big Square = Area of Small

Square + 4 Right Triangles

Now simplify the equation.

Simplify.

How does this equation apply to the right triangles in the figure?