Pythagorean Inequality Theorems

Learning Objectives State the Pythagorean Theorem, its converse, and the two Pythagorean Inequality Theorems. Determine if three given side lengths can form a valid triangle using the Triangle Inequality Theorem. Classify a triangle as acute, right, or obtuse given its three side lengths. Apply the Pythagorean Inequality Theorems to solve problems with unknown side lengths. Explain the geometric relationship between the square of the longest side and the sum of the squares of the other two sides for each triangle type. Construct a logical argument to justify the classification of a triangle. Ever tried to build a bookshelf and ended up with a wobbly, leaning mess? 📐 That's because the corners weren't perfect right angles! But how can you tell? You already know th...

TermDefinitionExample Converse of the Pythagorean TheoremThe reverse of the original theorem. It states that if the side lengths of a triangle satisfy the equation a² + b² = c², then the triangle must be a right triangle.If a triangle has sides of length 3, 4, and 5, we check if 3² + 4² = 5². Since 9 + 16 = 25, it is a right triangle. Acute TriangleA triangle where all three interior angles are less than 90 degrees.An equilateral triangle with side lengths 5, 5, and 5 is an acute triangle, as all its angles are 60°. Obtuse TriangleA triangle that has one interior angle greater than 90 degrees.A triangle with side lengths 4, 5, and 8 is an obtuse triangle because the angle opposite the side of length 8 is greater than 90°. Longest Side ('c')In the context of the Pythagorean Inequ...

Pythagorean Inequality: Acute Triangle If c^2 < a^2 + b^2, then the triangle is acute. Use this rule to prove a triangle is acute. If the square of the longest side ('c') is LESS THAN the sum of the squares of the other two sides, all angles are acute. Converse of the Pythagorean Theorem: Right Triangle If c^2 = a^2 + b^2, then the triangle is right. Use this rule to prove a triangle is right. If the square of the longest side ('c') is EQUAL TO the sum of the squares of the other two sides, the angle opposite side 'c' is a right angle. Pythagorean Inequality: Obtuse Triangle If c^2 > a^2 + b^2, then the triangle is obtuse. Use this rule to prove a triangle is obtuse. If the square of the longest side ('c') is GREATER THAN...