It looks like you are being asked to use the Pythagorean theorem to find the lengths of the sides of a right triangle. You've provided the following details:

• One side is 3 meters.
• Another side is 5 meters.
• The third side is $2m + 4$.

The Pythagorean theorem is given by:

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

where $a$ and $b$ are the two legs of the triangle, and $c$ is the hypotenuse (the longest side). We need to determine the value of $2m + 4$ by applying this relationship.

Let's check if the given values fit the Pythagorean theorem.

1. If the side of length 5 meters is the hypotenuse, then:

$3^2 + (2m + 4)^2 = 5^2$

Let's expand and solve for $m$.

$9 + (2m + 4)^2 = 25$

$(2m + 4)^2 = 16$

Taking the square root of both sides:

$2m + 4 = 4$

Solving for $m$:

$2m = 0 \quad \Rightarrow \quad m = 0$

So, the third side is:

$2(0) + 4 = 4 \, \text{meters}.$

Thus, the lengths of the sides of the triangle are:

• 3 meters
• 4 meters
• 5 meters

These form a well-known Pythagorean triple.

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Would you like a more detailed explanation of this? Let me know if you'd like me to go deeper into any steps!

Here are 5 related questions:

1. What are other Pythagorean triples and how are they formed?
2. How do you know which side is the hypotenuse in a right triangle?
3. Can the Pythagorean theorem be used for non-right triangles?
4. How do you apply the Pythagorean theorem to find the area of a right triangle?
5. How can you extend the Pythagorean theorem to 3-dimensional space?

Tip: Always verify the longest side in the triangle; it's the hypotenuse and always opposite the right angle.