Question
Consider a right-angled triangle with sides a=6
and c=9
. Determine whether these values form a Pythagorean triple. Show your steps and explain your answer.
in 2 to 3 sentences
and c=9
. Determine whether these values form a Pythagorean triple. Show your steps and explain your answer.
in 2 to 3 sentences
Answers
GPT-4o mini
In a right-angled triangle, the relationship between the sides is defined by the Pythagorean theorem, which states that \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse. Given \(a = 6\) and \(c = 9\), we can substitute these values:
\[
6^2 + b^2 = 9^2
\]
\[
36 + b^2 = 81
\]
\[
b^2 = 81 - 36 = 45
\]
Since \(b = \sqrt{45} = 3\sqrt{5}\), we conclude that \(6\), \(9\), and \(3\sqrt{5}\) do not form a Pythagorean triple because all sides must be whole numbers.
\[
6^2 + b^2 = 9^2
\]
\[
36 + b^2 = 81
\]
\[
b^2 = 81 - 36 = 45
\]
Since \(b = \sqrt{45} = 3\sqrt{5}\), we conclude that \(6\), \(9\), and \(3\sqrt{5}\) do not form a Pythagorean triple because all sides must be whole numbers.