5.2.3 The Pythagorean Theorem (continued) Example 4 Classifying Triangles
The Pythagorean Inequalities Theorem is used to determine whether a triangle is obtuse, acute, or right. When a triangle is a right triangle, the sum of the squares of its two legs, the shorter sides, are equal to the square of the hypotenuse. When a triangle is obtuse, the sum of the squares of the two shorter sides of the triangle is less than the square of the length of the longer side. When a triangle is acute, the sum of the squares of the shorter two sides in the triangle is greater than the square of the third side. The Triangle Inequality Theorem is used here to determine whether three measures can be the side lengths of a triangle. If it is a triangle, it is classified as acute, obtuse, or right. To determine if the measures can belong to the sides of a triangle, apply the Triangle Inequality Theorem. Sum the values in pairs and test whether the sum is greater than the third number. The values 5, 8, and 10 can be the sides of a triangle, since the sum of each pair is greater than the third number. Use the Pythagorean Inequalities Theorem to classify the triangle. The sum of the squares of the two shorter sides is 5 2 + 8 2 = 25 + 64 = 89. Since 89 is less than the square of the third side (10 2 = 100), the triangle is obtuse. In the second example, the sum of the two shorter measures is less than the third measure, so these values cannot be the lengths of a triangle.
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