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...
Answers
GPT-4o mini
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.
\[
a^2 + b^2 = c^2
\]
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.