Proving Pythagorean Theorem using Triangle Similarity Lesson Seed

G.SRT.4   Prove theorems about triangles.  Theorems include:  Pythagorean theorem proved using triangle similarity.

 

MP1:  Make sense of problems and perseveres in solving them.

MP2:  Reason abstractly and quantitatively.

 

In Grade 8, students are introduced to the Pythagorean theorem and explain a proof of the theorem using area models.  Students have also applied their understanding of Pythagorean theorem to solve real-world and mathematical problems.  Now, students will revisit the Pythagorean theorem to prove the theorem using triangle similarity.  Here are a few options for exploration of this proof:

1.      Geometer’s Sketchpad Investigation:  Either have students create similar right triangles using Sketchpad or have students use the SimilarRtTriangles.gsp file, and verify the triangles are similar using AA criterion.  Then pose the question, “If the three triangles are similar, what else can you verify?”  Allow students time to verify that the ratios of sides are proportional.  Next, label the sides of the large triangle “a,” “b,” and “c” and label the smaller segments of AB as c – x and x.  Have students work in groups to write equivalent ratios to prove that a2 +b2 = c2. (A video tutorial is available on the wiki.)  Have groups share strategies with the class.

2.      Patty Paper Investigation:  Have students create a right triangle using Patty Paper and find the altitude.  Have students label and verify that the triangles are similar. Then pose the question, “If the three triangles are similar, what else can you verify?”  Allow students time to measure and verify that the ratios of sides are proportional. Next, label the sides of the large triangle “a,” “b,” and “c” and label the smaller segments of AB as c – x and x.  Have students work in groups to write equivalent ratios to prove that a2 +b2 = c2Have groups share strategies with the class.

 

Extensions/Connections:

Select several of the proofs of the Pythagorean theorem and create a learning opportunity where students select which proof to master or students rotate between stations that develop a wider understanding of the proofs of the Pythagorean Theorem.  Examples of the variety of proofs of the Pythagorean Theorem can be found at the following web sites: 

 

http://www.jimloy.com/geometry/pythag.htm                                                              http://www.cut-the-knot.org/pythagoras/index.shtml     http://mathforum.org/mathtools/tool/25463/ http://jwilson.coe.uga.edu/emt668/emt668.student.folders/headangela/essay1/pythagorean.html http://www.mathsisfun.com/geometry/pythagorean-theorem-proof.html