A Proof of the Pythagorean Theorem


Activity #1

Proof of Pythagoras' Theorem.

  • Rotate the triangle in the applet below through 90 degrees using the “Rotate” slider. ​
  • Move the “Translate” slider to the right. ​
  • Follow the steps to find the area of the trapezoid(trapezium), and the area of the 3 triangles. ​
  • Equating them and simplifying, results to the proof of Pythagoras’ theorem.

Demonstrate Pythagoras’ Theorem.

  •  Moving the two blue points of the triangle change its size and shape. ​
  • Show the lengths and then check that Pythagoras’ theorem is correct. ​
  • The sliders demonstrate that the two small squares equal the big one.
  • First the square changes to a parallelogram and then to a rectangle. Try to figure out why the area of each shape stays the same.

Activity #2

Establish Pythagoras' Theorem.

  • Read the information below and complete the exercise that follows.

Both figures shown here below are squares with a side length of a + b. Notice that the first figure is divided into two squares and two rectangles. The second figure is divided into a square and four right triangles with legs of lengths a and b. Let’s call the hypotenuse of these triangles c.

(1.) What is the total area of each figure?

(2.) Find the area of each of the 9 smaller regions shown the figures and label them.

(3.) Add up the area of the four regions in Figure F and set this expression equal to the sum of the areas of the five regions in Figure G. If you rewrite this equation using as few terms as possible, what do you have?

Activity #3

Demonstration of Pythagoras' Theorem.

Use the applet to explore the relationship between areas. Follow the directions below.

  • Consider Squares A and B.
  • Check the box to see the area divided into five pieces with a pair of segments.
  • Check the box to see the pieces.
  • Arrange the five pieces to fit inside Square C.
  • Check the box to see the right triangle.
  • Arrange the figures so the squares are adjacent to the sides of the triangle.

Try it again with different squares. Estimate the areas of the new Squares, A, B, and C.

Estimate the areas of these new Squares, A, B, and C.

(1.) Explain what you observe as you complete the activity.

(2.) What do you think we may be able to conclude?

Challenge #1

Use the areas of the two identical squares to explain why 5² + 12² = 13².

Challenge #2

Find the unknown side lengths in these right triangles.

Challenge #3

Each number is between which two consecutive integers?

Quiz Time