We are given the following:

1. Outer square side length: 5 units (since the area of the outer square is 25 units, and the area of a square is given by $\text{side}^2$, so the side length is $\sqrt{25} = 5$ units).
2. The inner square cuts each side of the outer square into two segments, one 3 units long and the other 2 units long, meaning the inner square is positioned at an angle, not aligned with the sides of the outer square.
3. Sum of the areas of the 4 triangles: This is the difference between the area of the outer square and the area of the inner square.

Let’s break it down:

• Outer square area: $25 \, \text{units}^2$ (as mentioned).

• The four triangles are formed between the edges of the inner and outer square. Each side of the inner square cuts a side of the outer square into segments of 3 units and 2 units, creating right triangles.

• Sum of the areas of the 4 triangles = Area of the outer square - Area of the inner square.

To find the area of the inner square, we need to consider the geometry of the situation.

• Side length of the inner square: To calculate the side length of the inner square, observe that the inner square cuts the outer square's sides into 3 and 2 units, so the diagonal of the inner square equals the sum of the 3 and 2 units, i.e., $3 + 2 = 5$ units.

• For a square, the relationship between the diagonal $d$ and side length $s$ is given by the formula: $d = s\sqrt{2}$ Since the diagonal is 5 units, we can solve for $s$ (side length of the inner square): $5 = s\sqrt{2}$ $s = \frac{5}{\sqrt{2}} = \frac{5\sqrt{2}}{2} \approx 3.54 \, \text{units}$

• Area of the inner square = $s^2 = \left( \frac{5\sqrt{2}}{2} \right)^2 = \frac{25 \times 2}{4} = 12.5 \, \text{units}^2$.

• Sum of the areas of the 4 triangles = Area of the outer square - Area of the inner square: $25 - 12.5 = 12.5 \, \text{units}^2$

So, the sum of the areas of the 4 triangles is $\boxed{12.5} \, \text{units}^2$.

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Would you like further details or have any questions about the steps?

Here are some related questions to help expand your understanding:

1. How would the areas of the triangles change if the inner square was rotated more or less within the outer square?
2. Can you find the area of a triangle formed by two adjacent sides of the outer square and one side of the inner square?
3. What happens to the total area of the triangles if the inner square is placed inside the outer square without any rotation (aligned with the sides)?
4. How would the sum of the areas of the triangles change if the lengths of the segments created by the inner square on the outer square were different?
5. How could you apply the Pythagorean theorem to confirm the length of the side of the inner square?

Tip: Understanding how the inner square's diagonal relates to the side length is key to solving these types of problems.