7.2 Converse of Pythagorean Theorem

REMEMBER: The hypotenuse of a triangle is the longest side.

Theorems: Theorem 7.2: Converse of the Pythagorean Theorem: 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 2 sides, then the triangle is a RIGHT triangle. In General: If c2 = a2 + b2 then is a RIGHT triangle.

Theorems: Theorem 7.3: If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other 2 sides, then the triangle is an ACUTE triangle. In General: If c2 < a2 + b2 then is an ACUTE triangle.

Theorems: Theorem 7.4: If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other 2 sides, then the triangle is an OBTUSE triangle. In General: If c2 > a2 + b2 then is an OBTUSE triangle.

Example 1 Tell whether the triangle is a right triangle. . = 17.44 c = = 142 + 102 a = = b 304 = 196 + 100 304 = 296 Not a right triangle

Example 2 Tell whether the the given side lengths of a triangle can represent a right triangle a b c = 72 + 142 = 15.65 245 = 49 + 196 245 = 245 Yes, a right triangle

Example 3 a Decide if the segment lengths form a triangle. If so, would the triangle be acute, obtuse, or right? Must use Triangle Inequality Theorem to check the segments can make a triangle. = 39.34 Step 2: c2 ? a2 +b2 Step 1: ? 242 + 302 24 + 30 = 54 So 54 > 39.34 1548 ? 576 + 900 1548 ? 1476 so: 30 + 39.34 = 69.34 So 69.34 > 24 1548 > 1476 so: 24 + 39.34 = 63.34 So 63.34 > 30 The triangle is OBTUSE Is a triangle!

Example 3 b Decide if the segment lengths form a triangle. If so, would the triangle be acute, obtuse, or right? Must use Triangle Inequality Theorem to check the segments can make a triangle. Step 2: c2 ? a2 +b2 Step 1: 122 ? 82 + 102 8 + 10 = 18 So 18 > 12 144 ? 64 + 100 144 ? 164 so: 10 + 12 = 22 So 22 > 8 144 < 164 so: 8 + 12 = 20 So 20 > 10 The triangle is ACUTE Is a triangle!

Example 4 In order to be directly north, there must be a RIGHT angle! a = 305 ft b = 749 ft c = 800 ft 8002 = 3052 + 7492 640,000 = 654,026 No, not directly north because it does not form a right triangle.

Assignment Section 7.2 Pg 444; 1, 3-14, 16-21, 25-26, 29, 35, 36, 43