https://file.aiquickdraw.com/mathbot/file/1725490256308omohvol5.jpg

The image shows a geometric problem involving a hexagon \( ABCDEF \) which is constructed by juxtaposing four isosceles right triangles. The task is to calculate:

1. The perimeter of the hexagon.
2. The area of the hexagon.
3. The sum of the interior angles of the hexagon (this part asks only for a direct answer).

### Let's analyze and solve each part:

### 1. **Perimeter of the hexagon:**

From the image, it appears that the hexagon is composed of right triangles. Since the triangles are isosceles and right-angled, their sides are in a specific ratio: the legs (adjacent sides) are equal, and the hypotenuse is \( \sqrt{2} \times \text{leg} \).

To calculate the perimeter, we need the length of all sides of the hexagon. Let’s assume the base side \( AB \) is known as \( s \), we can use the given properties of the triangles to find the rest of the sides. The perimeter will be the sum of all six sides.

### 2. **Area of the hexagon:**

The area can be computed by breaking down the hexagon into the four right triangles that compose it. The area of one right triangle is given by:

\[
\text{Area of one triangle} = \frac{1}{2} \times \text{leg} \times \text{leg}
\]

Summing the areas of the four triangles will give the total area of the hexagon.

### 3. **Sum of the interior angles of the hexagon:**

For any polygon with \( n \) sides, the sum of the interior angles is given by the formula:

\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]

Since the hexagon has 6 sides, the sum of the interior angles is:

\[
(6 - 2) \times 180^\circ = 720^\circ
\]

This is the total sum of the interior angles.

---

Would you like me to provide further calculations for the perimeter and area? If yes, please provide any additional measurements or clarify the side lengths.

### Follow-up Questions:
1. Can you specify the length of any side of the hexagon or triangle?
2. Would you like a step-by-step breakdown of how to compute the perimeter and area?
3. Do you need a more detailed explanation of how the interior angle sum formula works?
4. Are you familiar with the properties of isosceles right triangles and their side ratios?
5. Do you need help understanding how polygons are constructed from triangles?

**Tip:** In any polygon, breaking the shape into smaller, simpler figures like triangles is an efficient way to calculate areas and perimeters.

Hexagon ABCDEF is constructed by juxtaposing four isosceles right triangles. Calculate: 1) The perimeter of the hexagon. 2) The area of the hexagon. 3) The sum of the interior angles of the hexagon.

Solve the problem of calculating the perimeter, area, and sum of interior angles for a hexagon constructed from four isosceles right triangles. This geometry problem covers key concepts in polygons and right triangle properties, suitable for Grade 8-10 students.

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