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
In a right-angled triangle, we can identify the sides based on their lengths. For a right triangle, the relationship between the lengths of the sides is given by the Pythagorean theorem, which states:
\[
a^2 + b^2 = c^2
\]
Where:
- \( c \) is the length of the hypotenuse (the longest side).
- \( a \) and \( b \) are the lengths of the other two sides.
In your case, you provided:
- \( a = 6 \) (one of the legs)
- \( c = 9 \) (the hypotenuse)
Since \( c \) is the longest side, let's denote \( b \) (the second leg) as unknown for now. According to the Pythagorean theorem, we can rearrange the equation to find \( b \):
\[
b^2 = c^2 - a^2
\]
Substituting the known values:
\[
b^2 = 9^2 - 6^2
\]
Calculating the squares:
\[
b^2 = 81 - 36
\]
\[
b^2 = 45
\]
Now we can find \( b \) by taking the square root of both sides:
\[
b = \sqrt{45} = 3\sqrt{5} \approx 6.71
\]
Now, we have the three sides:
- \( a = 6 \)
- \( b = 3\sqrt{5} \) (approximately 6.71)
- \( c = 9 \)
To determine if these values form a Pythagorean triple, we can check if \( a, b, c \) can be integers in the format \( (m^2 - n^2, 2mn, m^2 + n^2) \) for some integers \( m \) and \( n \). In this case, we can use the known sides:
Check if the integers \( 6 \), \( 9 \), and \( 3\sqrt{5} \) correspond to a known triple:
To check if these values form a Pythagorean triple, we must verify that:
\[
6^2 + (3\sqrt{5})^2 = 9^2
\]
Calculating each side:
- \( 6^2 = 36 \)
- \( (3\sqrt{5})^2 = 9 \times 5 = 45 \)
- \( 9^2 = 81 \)
Now, add the left side:
\[
36 + 45 = 81
\]
Both sides are equal, confirming the triangle's sides satisfy the Pythagorean theorem.
Since the side lengths satisfy the theorem, we conclude that \( 6 \), \( b = 3\sqrt{5} \), and \( 9 \) do not form a traditional Pythagorean triple with integer values. However, given [6 and 9] fits the Pythagorean relationship with the \( b \), we can say effectively this configuration forms a triangle.
In summary, the values given form a right triangle, demonstrating an application of the Pythagorean theorem, and although \( 3\sqrt{5} \) is not an integer, the other two sides meet the criteria of a right triangle. Thus, they do satisfy the requirement for a right triangle relationship.
\[
a^2 + b^2 = c^2
\]
Where:
- \( c \) is the length of the hypotenuse (the longest side).
- \( a \) and \( b \) are the lengths of the other two sides.
In your case, you provided:
- \( a = 6 \) (one of the legs)
- \( c = 9 \) (the hypotenuse)
Since \( c \) is the longest side, let's denote \( b \) (the second leg) as unknown for now. According to the Pythagorean theorem, we can rearrange the equation to find \( b \):
\[
b^2 = c^2 - a^2
\]
Substituting the known values:
\[
b^2 = 9^2 - 6^2
\]
Calculating the squares:
\[
b^2 = 81 - 36
\]
\[
b^2 = 45
\]
Now we can find \( b \) by taking the square root of both sides:
\[
b = \sqrt{45} = 3\sqrt{5} \approx 6.71
\]
Now, we have the three sides:
- \( a = 6 \)
- \( b = 3\sqrt{5} \) (approximately 6.71)
- \( c = 9 \)
To determine if these values form a Pythagorean triple, we can check if \( a, b, c \) can be integers in the format \( (m^2 - n^2, 2mn, m^2 + n^2) \) for some integers \( m \) and \( n \). In this case, we can use the known sides:
Check if the integers \( 6 \), \( 9 \), and \( 3\sqrt{5} \) correspond to a known triple:
To check if these values form a Pythagorean triple, we must verify that:
\[
6^2 + (3\sqrt{5})^2 = 9^2
\]
Calculating each side:
- \( 6^2 = 36 \)
- \( (3\sqrt{5})^2 = 9 \times 5 = 45 \)
- \( 9^2 = 81 \)
Now, add the left side:
\[
36 + 45 = 81
\]
Both sides are equal, confirming the triangle's sides satisfy the Pythagorean theorem.
Since the side lengths satisfy the theorem, we conclude that \( 6 \), \( b = 3\sqrt{5} \), and \( 9 \) do not form a traditional Pythagorean triple with integer values. However, given [6 and 9] fits the Pythagorean relationship with the \( b \), we can say effectively this configuration forms a triangle.
In summary, the values given form a right triangle, demonstrating an application of the Pythagorean theorem, and although \( 3\sqrt{5} \) is not an integer, the other two sides meet the criteria of a right triangle. Thus, they do satisfy the requirement for a right triangle relationship.