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 =
c2. 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
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under
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