Amsco Integrated Algebra I: The Pythagorean Theorem
 This is a free lesson from our course in Amsco's Integrated Algebra

 In this lesson you’ll learn concepts of the trigonometry of the right triangle. The word trigonometry is in broad sense means “measurement of triangles.” Here it covers applications to the learning of right triangles and Pythagorean Theorem. You may recall from earlier learning that a triangle that contains one and only one right angle i.e. 90 degree angle is called a right triangle. In this the side which is opposite the right angle, is called the hypotenuse and the other two sides of the triangle forming the right angle, are called the legs of the right triangle. (More text below video...)
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 (Continued from above) In relation to Algebra, the Pythagorean Theorem is an algebraic equation which relates the lengths of all the sides of a right triangle to each other. The theorem is one of the most important in trigonometry and helps to enhance the skills for solving real world problems. Pythagorean Theorem states that in a right triangle, the square of the length of hypotenuse is equal to the sum of the squares of the lengths of the other two sides i.e. each leg. If the length of the hypotenuse of right triangle ABC is represented by c and the lengths of the other two sides by a and b, the theorem of Pythagoras may be written as the formula: c2 = a2 + b2. Using this relationship, it is possible to solve a right triangle; or find the lengths of all the sides, when only two side lengths are given. To solve a right triangle using this theorem, you may substitute the two known side lengths into the equation, and then isolate the remaining unknown using addition or subtraction when necessary. Once isolated, unknown side length is determined by taking the square root of both sides. Solving for all the sides of a right triangle using Pythagorean Theorem will also enable further to solve for all of the angles, using trigonometry. For example: A 6m high straight bamboo stick was broken by the wind and the remaining section is still at a right angle to the ground. Its top touched the ground 2m from the root. If you are required to find out the lengths of the segments of the bamboo using Pythagorean Theorem, follow the below given procedure: • According to Pythagorean, for a right triangle with legs of length A and B, and hypotenuse of length C: C2 = A2 + B2. • Say A is the vertical section after it breaks, C be the broken section and B be the distance from the base to where the broken section touches the flat ground.    We then have two equations that describe what you know: A2 + 22 = C2   and          A + C = 6 You can solve by rearranging the second equation to solve i.e. A = 6 - C.    Then plug the value of A into the first equation: (6 - C) 2 + 4 = C2 36 - 12C + C2 + 4 = C2 40 - 12C = 0 40 = 12C C = 40/12 = 3.3m approx., as the final answer. Remember the following statements: • If a triangle is a right triangle, then the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. • If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, the triangle is a right triangle. The video above will help understand using several examples, more about the Pythagorean Theorem with right triangles.

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