The Pythagorean Theorem: Converse and Special Cases

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  • 0:01 The Pythagorean Theorem
  • 0:42 Law of Cosines
  • 1:48 Converse
  • 2:10 Using the Converse
  • 4:02 Lesson Summary
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Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

The Pythagorean Theorem is a famous theorem for right triangles. Watch this video to learn how the Pythagorean Theorem relates to the law of cosines and how the converse of the Pythagorean Theorem can help you identify right triangles.

The Pythagorean Theorem

The famous Pythagorean Theorem tells us that the square of the hypotenuse of a right triangle is always equal to the sum of the squares of the two sides. Written mathematically in formula form with sides a and b and hypotenuse c, it is a2 + b2 = c2. How can you remember this formula? You can think of a right triangle and then picture a square on each side of the triangle. Then you can tell yourself that you need to square each side and the two shorter sides will add up to the hypotenuse.

Law of Cosines

This Pythagorean Theorem is linked to the law of the cosines as a special case. All triangles that have one right angle are special cases of the law of cosines. The law of cosines is a2 + b2 - 2*a*b*cos (theta) = c2, where theta is the angle between the two sides a and b. When we have a right angle, cos (theta) becomes 0, and we get the Pythagorean Theorem.

a2 + b2 - 2*a*b*cos (theta) = c2

a2 + b2 - 2*a*b*0 = c2

a2 + b2 - 0 = c2

a2 + b2 = c2


We know that the Pythagorean Theorem tells us that right triangles will follow the formula a2 + b2 = c2. But the converse of this statement is also true. We can also say that if a triangle follows the formula a2 + b2 = c2, then the triangle is a right triangle.

Using the Converse

The converse of the Pythagorean Theorem is useful when we need to identify right triangles. Sometimes we want to know for sure that a particular triangle is a right triangle. We can apply the converse of the Pythagorean Theorem and see if the measurements of our triangle fit the formula. Let's see how we can do this.

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