To determine if a given triangle with sides \(a\), \(b\), and \(c\) is a right triangle using the Pythagorean theorem, we can use the converse of the Pythagorean theorem. The converse states that if the square of the length of the longest side \(c\) is equal to the sum of the squares of the other two sides \(a\) and \(b\), then the triangle is a right triangle.
-
Identify the lengths of the sides: Make sure to identify which of \(a\), \(b\), and \(c\) is the longest side since it will be the hypotenuse in a right triangle. We can assume \(c\) is the longest side without the loss of generality.
-
Check the Pythagorean theorem: The converse theorem states: \[ c^2 = a^2 + b^2 \] If this equation holds true, then the triangle with sides \(a\), \(b\), and \(c\) is a right triangle.
Here’s how to apply it step-by-step:
Example: Suppose \(a = 3\), \(b = 4\), and \(c = 5\).
-
Identify the sides: Here, \(c = 5\) (the longest side), \(a = 3\), and \(b = 4\).
-
Calculate: \[ c^2 = 5^2 = 25 \] \[ a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25 \]
-
Compare: Since \(c^2 = a^2 + b^2\) (\(25 = 25\)), according to the converse of the Pythagorean theorem, the triangle with sides 3, 4, and 5 is a right triangle.
If you provide specific values for \(a\), \(b\), and \(c\), I can help you determine whether that triangle is a right triangle using the same process!