area of inner square

Let's break this down step by step!

Step 1: Understanding the Situation

You have a larger square, and inside it, there is a smaller, tilted square (the inner square). The inner square intersects each side of the outer square, creating two segments of each side: one segment is 3 units long, and the other is 2 units long. So, each side of the outer square is divided into two parts: one of length 3 and the other of length 2.

This means the total length of each side of the outer square is:

$3 + 2 = 5 \text{ units}$

Step 2: Finding the Area of the Outer Square

The area of a square is given by the formula:

$\text{Area of a square} = (\text{side length})^2$

For the outer square, with side length 5 units, the area is:

$\text{Area of outer square} = 5^2 = 25 \text{ square units}$

Step 3: Area of the 4 Triangles

We need to determine the area of the four triangles that are formed between the inner square and the outer square. Each triangle is right-angled, and the two segments (3 units and 2 units) form the legs of the triangle. The area of one triangle is:

$\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height}$

For each triangle, the base is 3 units and the height is 2 units, so the area of one triangle is:

$\text{Area of one triangle} = \frac{1}{2} \times 3 \times 2 = 3 \text{ square units}$

Since there are 4 triangles, the total area of the triangles is:

$\text{Total area of 4 triangles} = 4 \times 3 = 12 \text{ square units}$

Step 4: Finding the Area of the Inner Square

Now, using the equation:

$\text{Area of outer square} - \text{Area of 4 triangles} = \text{Area of inner square}$

Substituting the known values:

$25 - 12 = 13$

So, the area of the inner square is 13 square units.

Final Answer:

$\text{Area of inner square} = 13 \text{ square units}$

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Would you like more details on how to approach these types of problems or any other questions?

Related Questions:

1. How do we calculate the area of a right-angled triangle?
2. What is the method to find the area of a square with a tilted position inside another square?
3. How would this approach change if the triangles were not right-angled?
4. If the side lengths of the outer square were different, how would the solution change?
5. How can you find the side length of a square given its area?

Tip: When solving problems involving shapes inside other shapes, it often helps to break the figure into simpler components, like triangles and squares, and work from there.