Using Similarity to Prove the Pythagorean Theorem

The Pythagorean Theorem is one of the most interesting theorems for two reasons: First, it’s very elementary; even high school students know it by heart. Second, it has hundreds of proofs. The proof below uses triangle similarity.

Pythagorean Theorem

In a right triangle with side lengths a and b and hypotenuse c,  the following equation always holds:

c^2 = a^2 + b^2. » Read more

Properties of Similar Triangles Part 2

This is the third and the conclusion of the Triangle Similarity Series. The two prequels  are 1. Introduction to Similarity and 2. Properties of Similar Triangles (Part 1).


In the previous post, we have investigated the properties of similar triangles. We have learned that corresponding angles of similar triangles are congruent. In this post, we are going to discuss more about the properties of similar triangles.  If you have not performed the investigation in the previous post, you can use the applet below.

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You would have realized from your exploration of the applet that aside from the angles, there is also something unique about the side lengths of the corresponding sides of the triangles (check the Show/Hide Side Length check box above).  We can verify they have the same ratio.  That is, if triangle ABC is similar to triangle DEF, then the following relationships hold: » Read more

Properties of Similar Triangles Part 1

This is the second part of the Triangle Similarity Series.  The first part is Introduction to Similarity.

In Introduction to Similarity, we have learned that similar objects have the same shape, but not necessarily have the same size. We drew a triangle using a graphics software zoomed it in and zoomed it out producing similar triangles.

Are Buu and Patrick 'similar'?

Zooming did not change the shape of the object. In effect, the measure of the interior angles of a triangle did not change.  » Read more

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