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8.MG.4a-e The student will apply the Pythagorean Theorem to solve problems involving right triangles, including those in context. Understanding the Standard The Pythagorean Theorem is essential for solving problems involving right triangles. The hypotenuse of a right triangle is the side opposite the right angle. The hypotenuse of a right triangle is always the longest side of the right triangle. The legs of a right triangle form the right angle. In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the legs. This relationship is known as the Pythagorean Theorem: a2+b2=c2. The Pythagorean Theorem is used to determine the measure of any one of the three sides of a right triangle when the measures of the other two sides are known. The converse of the Pythagorean Theorem states that if the square of the length of the hypotenuse equals the sum of the squares of the legs in a triangle, then the triangle is a right triangle. This can be used to determine whether a triangle is a right triangle given the measures of its three sides. The triangle inequality theorem states that the sum of any two sides of a triangle is greater than or equal to the third side. This theorem can be explored to check for reasonableness of solutions. Whole number triples that are the measures of the sides of right triangles, such as (3, 4, 5) and (5, 12, 13), are commonly known as Pythagorean triples. Additional sets of Pythagorean triples can be found by applying properties for similar triangles and proportional sides. For example, doubling the sides of a triangle with sides of (3, 4, 5) creates a Pythagorean triple of (6, 8, 10).

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