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 answer.

Answer 1

To determine if the sides $a=6$ and $c=9$ of a right-angled triangle form part of a Pythagorean triple, we first need to recognize the notation for a right-angled triangle, where:

• $a$ is one of the legs,
• $b$ is the other leg,
• $c$ is the hypotenuse (the longest side opposite the right angle).

In a Pythagorean triple, the sides of the triangle must satisfy the equation:

$$a^2+b^2=c^2$$

From the values given, we know:

• $a=6$
• $c=9$

Now we need to find $b$ and verify if there exists an integer $b$ such that the Pythagorean theorem holds.

1. Calculate $c^2$: $$c^2=9^2=81$$

2. Substitute $a$ into the Pythagorean theorem and solve for $b$: $$a^2+b^2=c^2$$ $$6^2+b^2=81$$ $$36+b^2=81$$

3. Rearranging the equation to solve for $b^2$: $$b^2=81-36$$ $$b^2=45$$

4. Taking the square root to find $b$: $$b=\sqrt{45}=\sqrt{9\cdot 5}=3\sqrt{5}$$ Since $3\sqrt{5}$ is not an integer, $b$ is not an integer.

5. Conclusion: The sides $a=6$, $b=3\sqrt{5}$, and $c=9$ do not form a Pythagorean triple because for a set of Pythagorean triples, all sides must be integers. Thus, $a=6$, $b=3\sqrt{5}$, and $c=9$ is not an integer solution.

Therefore, the values $6$ and $9$ cannot be considered parts of a Pythagorean triple as $b$ is not an integer.