To solve this problem, we use the Pythagorean theorem:

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

Where:

• $a$ and $b$ are the legs of the right triangle,
• $c$ is the hypotenuse.

From the diagram:

• $b = 12 \, \text{mm}$,
• $c = 13 \, \text{mm}$,
• $a$ is the missing side.

Step 1: Substitute the known values into the formula

$a^2 + 12^2 = 13^2$

Step 2: Simplify the squares

$a^2 + 144 = 169$

Step 3: Solve for $a^2$

$a^2 = 169 - 144$ $a^2 = 25$

Step 4: Take the square root

$a = \sqrt{25} = 5$

Thus, the missing leg is: $\boxed{5 \, \text{mm}}$

Would you like further clarification or details?

Related Questions:

1. How does the Pythagorean theorem apply to non-integer sides?
2. What happens when the hypotenuse is unknown in such problems?
3. Can this formula be used for non-right triangles? Why or why not?
4. How do you derive the formula for $a$ if $b$ and $c$ are given?
5. What are some real-life applications of the Pythagorean theorem?

Tip:

Always ensure the hypotenuse is the largest side when using the Pythagorean theorem to avoid mistakes.