5.2.3 The Pythagorean Theorem Key Objectives • Use the Pythagorean Theorem and its converse to solve problems. • Use Pythagorean inequalities to classify triangles. Key Terms • A set of three nonzero whole numbers a , b , and c such that a 2 + b 2 = c 2 is called a Pythagorean triple . Theorems, Postulates, Corollaries, and Properties • Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. • Converse of the Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. • Pythagorean Inequalities Theorem In △ ABC , c is the length of the longest side. If c 2 > a 2 + b 2 , then △ ABC is an obtuse triangle. If c 2 < a 2 + b 2 , then △ ABC is an acute triangle.
The Pythagorean Theorem is one of the most useful relationships in geometry. It states that in a right triangle, the sum of the squared side lengths of the legs of the triangle (the sides that include the right angle) is equal to the square of the length of the other side (the side opposite the right angle). The side opposite the right angle is called the hypotenuse. The Pythagorean Theorem can be used to find the length of one side of a right triangle when the lengths of the other two sides are known. The Converse of the Pythagorean Theorem states that if the sum of the squares of the two shorter sides of a triangle are equal to the square of the third angle, then that triangle is a right triangle. The Converse of the Pythagorean Theorem can be used to determine whether a triangle is a right triangle when the lengths of a triangle’s three sides are known.
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