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.
and c=9
. Determine whether these values form a Pythagorean triple. Show your steps and explain your answer.
Answers
GPT-4o mini
To determine if the values \(a = 6\), \(c = 9\), and \(b\) (the unknown side opposite the right angle) form a Pythagorean triple, we can use the Pythagorean theorem, which states:
\[
a^2 + b^2 = c^2
\]
Here, \(a\) is one leg of the triangle, \(b\) is the other leg, and \(c\) is the hypotenuse (the longest side). In this scenario, since \(c = 9\) is greater than \(a = 6\), we identify \(c\) as the hypotenuse.
To find \(b\), we reformulate the equation:
\[
b^2 = c^2 - a^2
\]
Now, we can substitute the known values into the equation:
1. Calculate \(a^2\):
\[
a^2 = 6^2 = 36
\]
2. Calculate \(c^2\):
\[
c^2 = 9^2 = 81
\]
3. Substitute into the equation for \(b^2\):
\[
b^2 = 81 - 36
\]
\[
b^2 = 45
\]
4. Calculate \(b\) by taking the square root of \(b^2\):
\[
b = \sqrt{45} = 3\sqrt{5}
\]
(Note that we only consider the positive value since side lengths cannot be negative.)
Now, we have the three sides:
- \(a = 6\)
- \(b = 3\sqrt{5}\)
- \(c = 9\)
To confirm that these sides form a Pythagorean triple, we check if the original Pythagorean theorem holds true:
\[
6^2 + (3\sqrt{5})^2 = 9^2
\]
Calculating both sides:
1. Left side:
\[
6^2 + (3\sqrt{5})^2 = 36 + 27 = 63
\]
2. Right side:
\[
9^2 = 81
\]
Since \(63 \neq 81\), we conclude that the values \(a = 6\) and \(c = 9\) with \(b = 3\sqrt{5}\) do not form a Pythagorean triple.
In summary, the sides \(a = 6\), \(c = 9\), with \(b = 3\sqrt{5}\) do not form a Pythagorean triple because they do not satisfy the Pythagorean theorem.
\[
a^2 + b^2 = c^2
\]
Here, \(a\) is one leg of the triangle, \(b\) is the other leg, and \(c\) is the hypotenuse (the longest side). In this scenario, since \(c = 9\) is greater than \(a = 6\), we identify \(c\) as the hypotenuse.
To find \(b\), we reformulate the equation:
\[
b^2 = c^2 - a^2
\]
Now, we can substitute the known values into the equation:
1. Calculate \(a^2\):
\[
a^2 = 6^2 = 36
\]
2. Calculate \(c^2\):
\[
c^2 = 9^2 = 81
\]
3. Substitute into the equation for \(b^2\):
\[
b^2 = 81 - 36
\]
\[
b^2 = 45
\]
4. Calculate \(b\) by taking the square root of \(b^2\):
\[
b = \sqrt{45} = 3\sqrt{5}
\]
(Note that we only consider the positive value since side lengths cannot be negative.)
Now, we have the three sides:
- \(a = 6\)
- \(b = 3\sqrt{5}\)
- \(c = 9\)
To confirm that these sides form a Pythagorean triple, we check if the original Pythagorean theorem holds true:
\[
6^2 + (3\sqrt{5})^2 = 9^2
\]
Calculating both sides:
1. Left side:
\[
6^2 + (3\sqrt{5})^2 = 36 + 27 = 63
\]
2. Right side:
\[
9^2 = 81
\]
Since \(63 \neq 81\), we conclude that the values \(a = 6\) and \(c = 9\) with \(b = 3\sqrt{5}\) do not form a Pythagorean triple.
In summary, the sides \(a = 6\), \(c = 9\), with \(b = 3\sqrt{5}\) do not form a Pythagorean triple because they do not satisfy the Pythagorean theorem.