The Pythagorean Theorem

This lesson consists of three different parts, any of which may be removed or expanded upon, depending on the amount of time available for the lesson.  Each part of the lesson applies the Pythagorean Theorem for mathematical understanding in some larger context.

One of the most important results of the Pythagorean Theorem is the distance formula.  Students can plot two moveable points in GSP, measure their coordinates, and calculate the distance between them as predicted by the Pythagorean Theorem.  They can then use GSP’s coordinate distance feature to see that their calculation and GSP’s calculation are the same.  GSP allows students to see many concrete examples of the distance formula, with points in quadrants I, II, III, and IV.  Students may also draw in the right triangle they use, and this triangle will change continuously as they move the two points.  In this way, students can see the distance formula in a general sense, recognizing that (x₁,y₁) and (x₂,y₂) are any two points.

Pythagorean triples are commonly used in construction.  For example, even thousands of years ago, the Egyptians used a 3-4-5 triangle made out of a 12-unit length of rope to construct right angles; today’s builders mark 6 feet and 8 feet from a corner on adjacent sides of a frame, then adjust the sides until the distance between the marks is 10 feet.  In the Excel activity, students can review their algebraic skills (e.g., Why do 2uv, u²-v², and u²+v² form a Pythagorean triple?) while using Excel’s formula feature to quickly and easily generate Pythagorean triples.

Finally, in the Other Shapes activity, students can extend the Pythagorean Theorem to shapes other than squares resting on the sides of a right triangle.  Different groups of students will work with different shapes (equilateral triangles, regular hexagons, and semicircles) to answer the following question: “Given two _________s of different sizes, construct a new _________ such that its area is equal to the sum of the two original shapes’ areas.”  GSP’s script tools make it easy for students to draw each of the given shapes, so drawing a diagram of their solution is simple.  In the semicircles activity, students can use GSP to extend the semicircle to a full circle, and can conjecture that the circle always goes through each of the triangle’s three vertices.  Thus, proving the fact that the midpoint of a right triangle’s hypotenuse is equidistant from the three vertices of the triangle is a possible extension of this activity.  In fact, applying the distance formula (see above) gives a straightforward proof of this theorem.  Students may also wonder if any shape placed on the edges of a right triangle would work; another extension is possible here.  Students will have to decide how they will determine the dimensions of a non-regular polygon (e.g., If we place rectangles on the sides, should the rectangles be of equal height, or of equal proportions?), and can extend their thinking on the previous problem to include this new, larger class of shapes.