Question

Consider a right-angled triangle with sides a = 6 and c = 9. Determine whether these values from a Pythagorean triple. show your steps and explain your explain your answer

Answer 1

To determine whether the given values \( a = 6 \) (one of the legs) and \( c = 9 \) (the hypotenuse) form a Pythagorean triple, we need to find the length of the other leg (let's call it \( b \)) and check if the relationship defined by the Pythagorean theorem holds. The Pythagorean theorem states that for a right-angled triangle:

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

### Step 1: Plug in the known values

We know:
- \( a = 6 \)
- \( c = 9 \)

We will solve for \( b \):

\[
6^2 + b^2 = 9^2
\]

Calculating the squares:

\[
36 + b^2 = 81
\]

### Step 2: Isolate \( b^2 \)

To find \( b^2 \), subtract \( 36 \) from both sides:

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

### Step 3: Solve for \( b \)

Now we take the square root of both sides to find \( b \):

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

### Step 4: Check if \( a, b, c \) form a Pythagorean triple

Now we will check if \( a = 6 \), \( b = 3\sqrt{5} \), and \( c = 9 \) can be expressed as integers.

However, a Pythagorean triple consists of three integer values. Since \( \sqrt{5} \) is not an integer, \( b = 3\sqrt{5} \) is not an integer, meaning that \( (a, b, c) = (6, 3\sqrt{5}, 9) \) does not form a Pythagorean triple.

### Conclusion

Therefore, the sides \( a = 6 \) and \( c = 9 \) do not form a Pythagorean triple because the resulting side \( b \) is not an integer. Thus, these values do not satisfy the condition for Pythagorean triples.