Visualizing the General Case of Pythagorean Theorem


Tapping It Up a Notch: Pythagorean Theorem – Part 2

Pythagorean Theorem - Segmenting The Area to Add to c-squared

In our last post, we used an inquiry/discovery approach to help students visualize how we can find the hypotenuse of a right-angle triangle when given the lengths of the two legs.

In this post, we will now introduce the General Case for Pythagorean Theorem in an attempt to use the same visual model to derive the formula for Pythagorean Theorem.

Using Pythagorean Theorem to Find the Length of the Hypotenuse

Visual Representation of Any Right Triangle [General Case]

In this video, we now introduce the concept for any right-angle triangle. Using the video from the previous post is probably a good idea as a starting point.

Summary of Pythagorean Theorem Video

Starting With The General Case Visually

We now start with variables representing any side lengths for a right-angle triangle.

Pythagorean Theorem - Visualization of the General Case

Connect Squaring Side-Lengths to Area

Once again, we show that squaring side lengths on the right-angle triangle will yield an area since we are multiplying length by width (or side by side, in this case):

Pythagorean Theorem - Spatial Reasoning - a-squared and b-squared

Segment the Area of the Shortest Leg into Pieces

In order to preserve the visualization of the sum of the squares of both legs being equivalent to the square of the length of the hypotenuse, we will need to chop up one of the areas into pieces:

Pythagorean Theorem - Segmenting The Area to Add to c-squared

Show The Sum of the Squares of the Leg Lengths Algebraically

As we did in the 3, 4, 5 case in the previous post, we now must use algebra to show what the sum of the squares of the leg lengths look like. This can help students start to see what the formula must be.

Pythagorean Theorem - Visually deriving the formula

Finding the Length of the Hypotenuse

In the previous case, finding the side length was pretty easy for students to do without necessarily consciously thinking about opposite operations. In this case, we must square-root the resulting area to find the length of the hypotenuse.

Pythagorean Theorem - Square Root Area of Hypotenuse to Get Length

Introduce c^2 as a Variable to Replace a^2 + b^2

Using substitution, we will replace a^2 + b^2 with a new variable, c^2.

Pythagorean Theorem - Square Root Area of Hypotenuse to Get Length

After students are comfortable with the algebra behind Pythagorean Theorem, you may choose to discuss this more deeply. Having students understand that we are actually substituting c with the square-root of a^2 + b^2 could be beneficial as they get closer to advanced functions and calculus.

Deriving the Formula for Pythagorean Theorem

Students may now be more comfortable understanding and using the algebraic representation of the Pythagorean Theorem Formula:

Pythagorean Theorem - Introduce the Algebraic Formula


Another post is forthcoming to help students better connect the visual representation of Pythagorean Theorem and the algebraic representation.

What do you think? How can we improve the introduction of this very important mathematical concept? Leave a comment below!


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3 comments on “Visualizing the General Case of Pythagorean Theorem

  1. David says:

    I like your general proof. You might be interested in this Sketchpad file I made to generalize this for the area of any regular shape used (instead of squares).
    http://mail.wecdsb.on.ca/~david_petro/sketches/PythagoreanAreaV5.gsp

    • Kyle Pearce Kyle Pearce says:

      Thanks!
      Very interested to check out the file. I am on my MacBook Pro and don’t have Geometer’s Sketchpad installed (probably should). Will check out on my iPad! Sounds like a pretty cool idea if I understand you correctly. Hope summer is treating you well!

    • Kyle Pearce Kyle Pearce says:

      Got a moment to check out your Sketchpad file and it is really cool! I think it is worth exploring in class to show students that the shape you choose is irrelevant, but a square is most efficient.

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