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 whether the sides \( a = 6 \), \( b \), and \( c = 9 \) form a Pythagorean triple, we need to check if they satisfy the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

1. Identify the sides:
- \( a = 6 \)
- \( c = 9 \)
- Since \( c \) is the longest side, it acts as the hypotenuse, and we need to find \( b \) (the other side).

2. According to the Pythagorean theorem, we have:
\[
c^2 = a^2 + b^2
\]
Plugging in the values we have:
\[
9^2 = 6^2 + b^2
\]
Calculating the squares:
\[
81 = 36 + b^2
\]

3. Solve for \( b^2 \):
\[
b^2 = 81 - 36
\]
\[
b^2 = 45
\]

4. Calculate \( b \):
\[
b = \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}
\]

Now we have:
- \( a = 6 \)
- \( b = 3\sqrt{5} \)
- \( c = 9 \)

To check if these sides form a Pythagorean triple, we need to check if \( a^2 + b^2 = c^2 \):

Calculate \( a^2 + b^2 \):
- \( a^2 = 6^2 = 36 \)
- \( b^2 = (3\sqrt{5})^2 = 9 \cdot 5 = 45 \)

Now, add them:
\[
a^2 + b^2 = 36 + 45 = 81
\]

Calculate \( c^2 \):
- \( c^2 = 9^2 = 81 \)

Since \( a^2 + b^2 = c^2 \) (both equal to 81), the sides do indeed form a Pythagorean triple.

**Conclusion:**
The sides \( 6 \), \( 3\sqrt{5} \), and \( 9 \) satisfy the Pythagorean theorem, and therefore these values do form a Pythagorean triple.