Raj made this argument: The Triangle Inequality Theorem states that if the lengths of three sides of a triangle are a, b, and c, then a + b > c. For this reason, since a, b, and c are all positive numbers, the square of a + b must be greater than the square of c. However, for a right triangle, the Pythagorean Theorem states that if the lengths of the legs are a and b, and if the length of the hypotenuse is c, then a2 + b2 = c2, not a2 + b2 > c2. Therefore, since the Pythagorean Theorem is known to be correct, and since right triangles should adhere to the Triangle Inequality Theorem, the Triangle Inequality Theorem must be incorrect. What is wrong with Raj's argument?

Question

Mathematics

Ariel Clements

11 October, 04:55

Raj made this argument: The Triangle Inequality Theorem states that if the lengths of three sides of a triangle are a, b, and c, then a + b > c. For this reason, since a, b, and c are all positive numbers, the square of a + b must be greater than the square of c. However, for a right triangle, the Pythagorean Theorem states that if the lengths of the legs are a and b, and if the length of the hypotenuse is c, then a2 + b2 = c2, not a2 + b2 > c2. Therefore, since the Pythagorean Theorem is known to be correct, and since right triangles should adhere to the Triangle Inequality Theorem, the Triangle Inequality Theorem must be incorrect. What is wrong with Raj's argument?

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Answers (1)

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Francesca Holloway · Answer 1

He is wrong because if a + b > c it doesn't follow that a^2 + b^2 > c^2

(a + b^2 will always be greater than c^2 but a^2 + b^2 will not always be.

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An example:

Right angled triangle with sides 3, 4 and 5:-

3 + 4 > 5 but 3^2 + 4^2 = 5^2