Pythagoras, who lived around 500 BCE is credited with being the first to prove that for all right triangles with side lengths a, b and c, [math]c^2=a^2+b^2[/math]. In this activity, you will prove that this statement is indeed true for all right triangles.[br][br]For this task, you will be exploring one visual proof of the Pythagorean Theorem discovered by Bhaskara in the 12th century.
[b][color=#0000ff][size=200]Warm Up[/size][/color][/b]
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.
[b][color=#0000ff][size=200]Part 1[/size][/color][/b]
Drag and drop the triangles until the vertices touch.[br][br][i]Hint: If you aren't sure what shape to make, zoom out and use the shape I placed as your guide![/i]
Prove that the shape formed actually is a square by nothing the characteristics of the angles and side lengths.[br][br]"The angles are ______ and the side lengths are ______."
The angles are all 90 degrees and the side lengths are all congruent (a+b).
We already know that the angle across from the hypotenuse [math]c[/math] is a right angle. [br][br]- Place a red dot at each of the [i]right[/i] angles located across from side length [math]c[/math].[br][br]- Place a blue dot at each of the [i]acute[/i] angles located across from side length [math]a[/math].[br][br]- Place a green dot at each of the [i]acute[/i] angles located across from side length [math]b[/math].
State the sum of the angles for any triangle.
Now determine the sum of one blue angle and one green angle.[br][br][i]Hint: you already know the red angles are 90 degrees[/i]
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 [math]c[/math].[br][br][i]Hint: you showed above that green + blue = 90. So what must be left to form a straight line?[/i]
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.[br][br]"The angles are ______ and the side lengths are ______."
The angles are all 90 degrees and the side lengths are all congruent (c).
To find the area of this square, we can find the sum of the areas of each of the five polygons.[br][br]We already found the area of one triangle in the first question ([math]\frac{1}{2}ab[/math]).[br]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.
[color=#ff0000][b][size=150][size=200]STOP! [br][/size][/size][/b][/color]Check your answer with a partner before moving on to the next step in this proof.
[b][color=#0000ff][size=200]Part 2[/size][/color][/b]
Here is a new square with the same side lengths and, therefore, the same area.
Find the area of each individual rectangle. [br][br][i]Hint: I encourage you to write the area of each rectangle directly on the image.[/i][br][br]Then add them together and combine like terms.
[color=#ff0000][b][size=150][size=200]STOP! [br][/size][/size][/b][/color]Check your answer with a partner before moving on to the next step in this proof.
[b][color=#0000ff][size=200]Part 3[/size][/color][/b][br][br]Now that you've found the areas of each large square, you're ready to prove the Pythagorean Theorem!
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.[br][br]____________ = ____________[br] (first area) (second area)
[math]2ab+c^2=a^2+2ab+b^2[/math]
Which term do you see that appears on [b]both[/b] sides of the equation?
Subtract that common term from both sides of the equation.[br][br]Write what you are left with.
[b][color=#0000ff][size=200]Reflection[/size][/color][/b]
Explain in your own words how this demonstration proves the Pythagorean Theorem.
Make note of any questions you still have about this proof.