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 given sides \( a = 6 \) and \( c = 9 \) form a Pythagorean triple in the context of a right-angled triangle, we need to identify a potential third side, \( b \), and check if the Pythagorean theorem holds. The Pythagorean theorem states:

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

In this equation:

- \( a \) is one leg of the triangle.
- \( b \) is the other leg of the triangle.
- \( c \) is the hypotenuse (the longest side).

We can first identify which side is the hypotenuse. Since \( c = 9 \) is greater than \( a = 6 \), we can use \( c \) as the hypotenuse and denote \( a \) and \( b \) as the legs.

Now, we need to find \( b \) and check if the relationship holds. Since \( a \) and \( c \) are given:

1. We know \( c = 9 \).
2. We also know \( a = 6 \).

Now, we can rearrange the Pythagorean theorem to solve for \( b \):

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

Calculating \( c^2 \) and \( a^2 \):

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

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

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

To find the value of \( b \):

\[
b = \sqrt{45} \approx 6.71
\]

Now we have:

- \( a = 6 \)
- \( b \approx 6.71 \)
- \( c = 9 \)

Next, we check if \( a, b, c \) form a Pythagorean triple by verifying:

\[
6^2 + (\sqrt{45})^2 = 9^2
\]

Calculating both sides:

Left-hand side:

\[
6^2 + 45 = 36 + 45 = 81
\]

Right-hand side:

\[
9^2 = 81
\]

Both sides are equal, confirming that \( 6^2 + (\sqrt{45})^2 = 9^2 \) holds true.

Thus, \( a = 6 \), \( b \approx 6.71 \), and \( c = 9 \) indeed follow the Pythagorean theorem, and we conclude:

**Answer**: Yes, these values form a Pythagorean triple with \( a = 6 \), \( b \approx 6.71 \), and \( c = 9 \).