Proving Pythagoras Right
First we'll start with analyzing one right triangle.
Determine the algebraic expression that represents the area of the triangle below.
Imagine if we were to copy the triangle, rotate it clockwise, and translate it so just the vertices were touching; then repeat the process two more times. Predict what shape would be formed.
The right triangle will be copied and rotated three times to form a larger polygon.
Prove that the shape formed actually is a square by nothing the characteristics of the angles and side lengths. "The angles are ______ and the side lengths are ______."
The polygon at the center of the figure appears to be a square. Let's prove it!
State the sum of the angles for any triangle.
Now determine the sum of one blue angle and one green angle. Hint: you already know the red angles are 90 degrees
State the sum of the angles of any straight line.
Now determine the sum of each angle in the polygon located in the center of the green figure with side lengths . Hint: you showed above that green + blue = 90. So what must be left to form a straight line?
Summarize how you know that the polygon in the center of the figure is a square. Note the characteristics of the angles and side lengths. "The angles are ______ and the side lengths are ______."
Now that that's all squared away (pun intended), let's find the area of this big square!
To find the area of this square, we can find the sum of the areas of each of the five polygons. We already found the area of one triangle in the first question (). Multiply this expression by 4 to determine the area of all four triangles. Simplify your answer.
Find the area of the center (green) square. Simplify your expression.
Add these two values together to get the total area of the large square.
Note that the length of each side of the original square was a+b.
Find the area of each individual rectangle. Hint: I encourage you to write the area of each rectangle directly on the image. Then add them together and combine like terms.
Knowing that the areas of the two squares are equal, write an equation with the area of the first large square (made of triangles) equal to the area of the second large square that you just found. ____________ = ____________ (first area) (second area)
Which term do you see that appears on both sides of the equation?
Subtract that common term from both sides of the equation. Write what you are left with.
Explain in your own words how this demonstration proves the Pythagorean Theorem.
Make note of any questions you still have about this proof.