PYTHAGOREAN THEOREM In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs. Thus, in Fig. 1.1, c2 = a2 + b2 Fig. 1.1 Tests for Right, Acute, and Obtuse Triangles If c2 = a2 + b2 applies to the three sides of a triangle, then the triangle is a right triangle; but if c2 ≠ a2 + b2, then the triangle is not a right triangle. In ABC, if c2 < a2 + b2 where c is the longest side of the triangle, then the triangle is an acute triangle. Thus in Fig. 1.2, 92 < 62 + 82 (that is, 81 < 100); hence, ABC is an acute triangle. In ABC, if c2 > a2 + b2 where c is the longest side of the triangle, then the triangle is an obtuse triangle. Thus in Fig. 1.3, 112 > 62 + 82 (that is, 121 > 100); hence, ABC is an obtuse triangle. Fig. 1.2 Fig. 1.3 Finding the sides of a right triangle If leg a is the missing side, then transform the equation to the form: a = √(𝑐 2 − 𝑏 2 ) If leg b is unknown, then: b = √(𝑐 2 − 𝑎2 ) For hypotenuse c missing, then: c = √(𝑎2 + 𝑏 2 )