The problem presents a right-angled triangle, and the goal is to find the value that should go in the box, which likely refers to a missing dimension.

Here's the process to solve this:

1. Given Information:

   • The hypotenuse of the triangle is 12 cm.
   • The height perpendicular to the base (from the right angle) is 2 cm.
   • The other leg of the triangle is 3 cm.

2. Finding the Missing Length (Box Value): Since the triangle has one leg given (3 cm), we need to determine the length of the other leg (which is probably what should go in the box). For a right-angled triangle, we can apply the Pythagorean theorem:

   $\text{Hypotenuse}^2 = \text{Leg}_1^2 + \text{Leg}_2^2$

   Let's assign the variables:

   • Hypotenuse = 12 cm
   • Leg 1 = 3 cm (the vertical leg)
   • Leg 2 = ? (the horizontal leg we are solving for)

3. Solve Using the Pythagorean Theorem:

   $12^2 = 3^2 + \text{Leg}_2^2$

   $144 = 9 + \text{Leg}_2^2$

   $\text{Leg}_2^2 = 144 - 9 = 135$

   $\text{Leg}_2 = \sqrt{135} \approx 11.62 \, \text{cm}$

Thus, the missing value for the box should be approximately 11.62 cm.

Would you like more details, or do you have any questions?

Here are 5 related questions to explore:

1. How does the Pythagorean theorem work for different types of right-angled triangles?
2. What are the different ways to solve for unknown sides in a triangle?
3. How can you calculate the area of this triangle?
4. How does the height (2 cm) affect the area of the triangle?
5. How can you verify the accuracy of triangle dimensions when given inaccurate drawings?

Tip: Always check the triangle's proportions before assuming it's drawn to scale, as drawings might not always be accurate.