Question

1. which conditions imply a triangle is right - angled based on the pythagorean theorem?

a. $a^{2}+b^{2}>c^{2}$
b. $a + b = c$
c. $a^{2}+b^{2}=c^{2}$
d. $a^{2}+b^{2}

Explanation:

Step1: Recall Pythagorean Theorem

In a right - angled triangle, if \(a\) and \(b\) are the lengths of the two legs and \(c\) is the length of the hypotenuse, then \(a^{2}+b^{2}=c^{2}\).

Step2: Analyze each option

• Option a: \(a^{2}+b^{2}>c^{2}\) implies an acute - angled triangle (by the converse of the Pythagorean theorem).
• Option b: \(a + b=c\) does not form a triangle (triangle inequality \(a + b>c\) for non - degenerate triangles).
• Option c: \(a^{2}+b^{2}=c^{2}\) is the exact statement of the Pythagorean theorem for a right - angled triangle.
• Option d: \(a^{2}+b^{2}

Answer:

C. \(a^{2}+b^{2}=c^{2}\)