Question

There are sides 9 units^2, 36 units^2, and 64 units^2. Is this proof of the Pythagorean Theorem? Explain

Answer 1

To determine if the given sides (9 units², 36 units², and 64 units²) can be used to prove the Pythagorean Theorem, we first need to recognize that the Pythagorean Theorem applies to the lengths of the sides of a right triangle, not the areas of the sides.

According to the Pythagorean Theorem, for a right triangle with legs of lengths \( a \) and \( b \), and hypotenuse \( c \), the relationship is given by:

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

We need to extract the lengths from the areas you've given. The areas are:

1. Side corresponding to an area of 9 units²: \(\sqrt{9} = 3\) units
2. Side corresponding to an area of 36 units²: \(\sqrt{36} = 6\) units
3. Side corresponding to an area of 64 units²: \(\sqrt{64} = 8\) units

Now we have the triangle sides as 3, 6, and 8. We then need to verify if they satisfy the Pythagorean Theorem:

We'll check if:

\[
3^2 + 6^2 = 8^2
\]

Calculating each side:

- \(3^2 = 9\)
- \(6^2 = 36\)
- \(8^2 = 64\)

Now summing the squares of the two shorter sides:

\[
9 + 36 = 45
\]

Since \(8^2 = 64\), we see that:

\[
45 \neq 64
\]

Thus, the relationship does not hold. Given that the sides do not fit the criterion set by the Pythagorean Theorem, we conclude that these three areas (and the corresponding lengths) do not provide a proof of the Pythagorean theorem. Therefore, they do not form a right triangle.