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 given sides \( a = 6 \) and \( c = 9 \) can form a right-angled triangle, we need to find the third side \( b \). In a right-angled triangle, the Pythagorean theorem states:
\[
a^2 + b^2 = c^2
\]
Where \( a \) and \( b \) are the lengths of the two legs (the sides that form the right angle), and \( c \) is the length of the hypotenuse (the side opposite the right angle).
In this case, we have:
- \( a = 6 \)
- \( c = 9 \)
Now, we need to find if there exists a \( b \) such that:
\[
6^2 + b^2 = 9^2
\]
Calculating \( 6^2 \) and \( 9^2 \):
\[
6^2 = 36
\]
\[
9^2 = 81
\]
Now substituting these values back into the equation:
\[
36 + b^2 = 81
\]
To isolate \( b^2 \), we perform the following steps:
1. Subtract \( 36 \) from both sides:
\[
b^2 = 81 - 36
\]
\[
b^2 = 45
\]
2. Now take the square root of both sides to find \( b \):
\[
b = \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}
\]
Now we have found \( b \). The lengths of the triangle are:
- \( a = 6 \)
- \( b = 3\sqrt{5} \)
- \( c = 9 \)
Next, we check if these sides can form a Pythagorean triple by checking if they satisfy the equality \( a^2 + b^2 = c^2 \). We already established this from our calculations:
- \( a^2 = 36 \)
- \( b^2 = 45 \)
- \( c^2 = 81 \)
Verifying the Pythagorean theorem:
\[
36 + 45 = 81
\]
Since \( 81 = 81 \), the sides \( 6 \), \( 3\sqrt{5} \), and \( 9 \) indeed satisfy the Pythagorean theorem.
### Conclusion:
Yes, the values of \( a = 6 \) and \( c = 9 \) (with \( b = 3\sqrt{5} \)) can form a right-angled triangle, and they do form a Pythagorean triple. The sides fulfilling the Pythagorean theorem are \( 6 \), \( 3\sqrt{5} \), and \( 9 \). However, it is important to note that in the context of whole number lengths, \( (6, 8, 10) \) would be a more conventional example of a Pythagorean triple.
\[
a^2 + b^2 = c^2
\]
Where \( a \) and \( b \) are the lengths of the two legs (the sides that form the right angle), and \( c \) is the length of the hypotenuse (the side opposite the right angle).
In this case, we have:
- \( a = 6 \)
- \( c = 9 \)
Now, we need to find if there exists a \( b \) such that:
\[
6^2 + b^2 = 9^2
\]
Calculating \( 6^2 \) and \( 9^2 \):
\[
6^2 = 36
\]
\[
9^2 = 81
\]
Now substituting these values back into the equation:
\[
36 + b^2 = 81
\]
To isolate \( b^2 \), we perform the following steps:
1. Subtract \( 36 \) from both sides:
\[
b^2 = 81 - 36
\]
\[
b^2 = 45
\]
2. Now take the square root of both sides to find \( b \):
\[
b = \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}
\]
Now we have found \( b \). The lengths of the triangle are:
- \( a = 6 \)
- \( b = 3\sqrt{5} \)
- \( c = 9 \)
Next, we check if these sides can form a Pythagorean triple by checking if they satisfy the equality \( a^2 + b^2 = c^2 \). We already established this from our calculations:
- \( a^2 = 36 \)
- \( b^2 = 45 \)
- \( c^2 = 81 \)
Verifying the Pythagorean theorem:
\[
36 + 45 = 81
\]
Since \( 81 = 81 \), the sides \( 6 \), \( 3\sqrt{5} \), and \( 9 \) indeed satisfy the Pythagorean theorem.
### Conclusion:
Yes, the values of \( a = 6 \) and \( c = 9 \) (with \( b = 3\sqrt{5} \)) can form a right-angled triangle, and they do form a Pythagorean triple. The sides fulfilling the Pythagorean theorem are \( 6 \), \( 3\sqrt{5} \), and \( 9 \). However, it is important to note that in the context of whole number lengths, \( (6, 8, 10) \) would be a more conventional example of a Pythagorean triple.