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. (2 points)

Answer 1

To determine if the sides \(a = 6\) and \(c = 9\) of a right-angled triangle form a Pythagorean triple, we first need to identify which sides correspond to the definitions of a Pythagorean triple. In a right-angled triangle, the Pythagorean theorem states:

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

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

Given:
- \(a = 6\) (one of the legs),
- \(c = 9\) (the hypotenuse since it is the longest side).

Next, we need to determine the value of the other leg \(b\). We can reformulate the Pythagorean theorem to solve for \(b\):

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

Now, let's calculate \(c^2\) and \(a^2\):

\[
c^2 = 9^2 = 81
\]
\[
a^2 = 6^2 = 36
\]

Now, substituting these values into the equation for \(b^2\):

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

To find \(b\), we take the square root of \(45\):

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

Now we have the sides of the triangle as \(a = 6\), \(b = 3\sqrt{5}\), and \(c = 9\).

To check if these values form a Pythagorean triple, we need to verify if:

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

Calculating \(b^2\):

\[
b^2 = (3\sqrt{5})^2 = 9 \times 5 = 45
\]

Now check the Pythagorean theorem:

\[
a^2 + b^2 = 36 + 45 = 81
\]
\[
c^2 = 81
\]

Since \(c^2 = a^2 + b^2\) holds true, the values do indeed form a Pythagorean triple.

### Conclusion:

The sides \(a = 6\), \(b = 3\sqrt{5}\), and \(c = 9\) satisfy the Pythagorean theorem, confirming that they form a Pythagorean triple. Thus, the answer is **yes**, these values form a Pythagorean triple.