**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.
Geometers
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 *a ^{2} +b^{2}
= c^{2}. *(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 *a ^{2} +b^{2} = c^{2}. *Have 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