Understanding the Pythagorean Relationship

  • 6.5 Understanding the Pythagorean Relationship

    Grade: 6th to 8th

    This e-example provides a proof without words for the Pythagorean Theorem.



    The red and yellow squares on the legs of the triangle can be transformed into parallelograms and then into rectangles inside the square attached to the hypotenuse. Use the sliders to accomplish this.
    You can also change the size of the right triangle by clicking and dragging a vertex on the triangle (a green dot will appear if done correctly).


    Your goal is to determine how the interactive figure demonstrates the Pythagorean relationship. Consider the right triangle in the interactive figure. Red and yellow squares have been constructed on its legs. A blue square has also been constructed on its hypotenuse. How does your transformation of the squares into parallelograms and then into rectangles affect their area? What relationship is demonstrated when the rectangles fill the large square formed on the hypotenuse? Now drag the red and yellow rectangles back to their original positions as squares on the legs of the triangle.
    Change the size of the triangle and repeat the process describes above. What do you observe?

    Objectives and Standards

    NCTM Standards and Expectations
    • Geometry / Measurement
    • Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
    • 6-8
    • Geometry