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Geometry: High School15 chapters | 160 lessons

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Lesson Transcript

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 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 *a*2 + *b*2 = *c*2. 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.

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 *a*2 + *b*2 - 2**a***b**cos (theta) = *c*2, 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.

*a*2 + *b*2 - 2**a***b**cos (theta) = *c*2

*a*2 + *b*2 - 2**a***b**0 = *c*2

*a*2 + *b*2 - 0 = *c*2

*a*2 + *b*2 = *c*2

We know that the Pythagorean Theorem tells us that right triangles will follow the formula *a*2 + *b*2 = *c*2. But the converse of this statement is also true. We can also say that if a triangle follows the formula *a*2 + *b*2 = *c*2, then the triangle is a right triangle.

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.

Let's say you are an architect, and your boss needs you to figure out if a particular triangle in a building project is a right triangle or not. You can apply the converse of the Pythagorean Theorem to solve the problem. You ask your boss for the measurements of the sides first so that you have all the data you need. Then you write down the formula for the Pythagorean Theorem. You remember it to be *a*2 + *b*2 = *c*2. Then you tell yourself that the converse is true; that if the numbers fit the formula, then the triangle must be a right triangle. So, your boss has given you 2.2 and 4.4 for the two sides and square root (24.2) for the hypotenuse. Now, what you need to do is to plug these numbers into the formula to check.

2.22 + 4.42 = (square root (24.2))2

4.84 + 19.36 = 24.2

24.2 = 24.2

It looks like the calculation is a success and the triangle in question is indeed a right triangle. You report this back to your boss, telling him that it is okay to proceed with the project.

What have we learned? We've learned that the **Pythagorean Theorem** tells us that *for all right triangles, the square of the hypotenuse is always equal to the sum of the squares of the two other sides*. The Pythagorean Theorem is a special case of the law of cosines, *a*2 + *b*2 - 2**a***b**cos (theta) = *c*2 because cos (theta) = 0 when the angle is a 90 degree or right angle. We also learned that the converse of the Pythagorean Theorem is true as well. We can say that if a triangle's sides fit the formula *a*2 + *b*2 = *c*2, then the triangle must be a right triangle. We can use this information to help us identify right triangles.

Explore this video lesson's information if your objectives are to:

- State the Pythagorean Theorem
- Write and remember the formula for the Pythagorean Theorem
- Impart your knowledge that the Pythagorean Theorem is a special case of the law of cosines
- Determine whether a triangle is a right triangle when you know the measurements of the sides

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Geometry: High School15 chapters | 160 lessons

- Ratios and Proportions: Definition and Examples 5:17
- Geometric Mean: Definition and Formula 5:15
- Angle Bisector Theorem: Definition and Example 4:58
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- Constructing Similar Polygons 4:59
- Properties of Right Triangles: Theorems & Proofs 5:58
- The Pythagorean Theorem: Practice and Application 7:33
- The Pythagorean Theorem: Converse and Special Cases 5:02
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