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...

Answer 1

To determine whether the values \(a = 6\) and \(c = 9\) form a Pythagorean triple, we first need to identify what a Pythagorean triple is. A Pythagorean triple consists of three positive integers \(a\), \(b\), and \(c\) such that:

\[
a^2 + b^2 = c^2
\]

where \(c\) is the length of the hypotenuse (the longest side) of the right-angled triangle, and \(a\) and \(b\) are the lengths of the other two sides.

In this scenario, we have the following values:
- Side \(a = 6\)
- Hypotenuse \(c = 9\)

Now, we need to find the value of \(b\) (the other side) and check whether the Pythagorean theorem holds true.

1. First, we calculate \(c^2\):

\[
c^2 = 9^2 = 81
\]

2. Next, we calculate \(a^2\):

\[
a^2 = 6^2 = 36
\]

3. Now, we rearrange the Pythagorean theorem to find \(b^2\):

\[
b^2 = c^2 - a^2
\]

Substituting in the values we calculated:

\[
b^2 = 81 - 36 = 45
\]

4. We then calculate the value of \(b\):

\[
b = \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}
\]

5. Now, we have our sides:
- \(a = 6\)
- \(b = 3\sqrt{5} \approx 6.708\)
- \(c = 9\)

To check for a Pythagorean triple, we need to verify if the sides can all be integers. Since \(b\) is not an integer, the sides \(a = 6\) and \(c = 9\) do not form a Pythagorean triple.

**Conclusion:** The values \(a = 6\), \(b = 3\sqrt{5}\), and \(c = 9\) do not form a Pythagorean triple because \(b\) is not an integer.