https://file.aiquickdraw.com/mathbot/file/17266424539648887msw5.jpg

Let's work through the questions one by one based on the given figures and the data provided:

### i) What is the length of the common side of triangles ΔAFD and ΔFED?

The common side between triangles ΔAFD and ΔFED is side **FD**. From **Fig. II**, we can see that **FD = 4 cm** (since the dimension is explicitly given).

Thus, the length of the common side FD is **4 cm**.

### ii) If DE = 11 cm, find the semi-perimeter of ΔFED.

The semi-perimeter (s) of a triangle is given by the formula:

\[
s = \frac{a + b + c}{2}
\]

In ΔFED, the sides are:
- **FD = 4 cm**
- **DE = 11 cm** (given)
- **FE = 3 cm** (from Fig. II)

So, the semi-perimeter \( s \) is:

\[
s = \frac{4 + 11 + 3}{2} = \frac{18}{2} = 9 \, \text{cm}
\]

Thus, the semi-perimeter of ΔFED is **9 cm**.

### iii) Find the area of ΔFED.

The area of a triangle can be calculated using Heron’s formula, which is:

\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\]

where \( s \) is the semi-perimeter, and \( a \), \( b \), and \( c \) are the sides of the triangle.

For ΔFED:
- \( s = 9 \, \text{cm} \) (as calculated above)
- \( a = 4 \, \text{cm} \)
- \( b = 11 \, \text{cm} \)
- \( c = 3 \, \text{cm} \)

Now applying Heron's formula:

\[
\text{Area} = \sqrt{9(9 - 4)(9 - 11)(9 - 3)} = \sqrt{9 \times 5 \times (-2) \times 6}
\]

It seems there is an issue here, as Heron's formula gives a non-real result with this set of side lengths. Since one of the sides may not form a valid triangle, we might need to recheck the triangle's dimensions. There may be an error in the input data, or the dimensions provided do not allow for a valid triangle.

### Alternative: If you wish to solve the third option instead:

#### iii) What is the ratio of the area of ΔAFD and ΔDCE?

To find the ratio of the areas of ΔAFD and ΔDCE, we would first calculate the area of each triangle. For now, based on their similarity in base and height comparison:

- Both triangles have the same height (8 cm).
- The bases of ΔAFD and ΔDCE are proportional (length of FD and length of DC).

However, to give a precise ratio, we'd calculate the actual areas. Let me know if you'd prefer to go with this question instead.

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

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Here are some related questions to consider:
1. What is the perimeter of triangle ΔAFD?
2. How would you find the height of ΔFED using basic trigonometry?
3. What are the properties of triangles that don't satisfy Heron's formula?
4. Can you calculate the area of ΔDCE directly from the given data?
5. What is the relationship between the areas of the triangles formed within the rectangle?

**Tip**: When applying Heron's formula, always verify that the sides provided form a valid triangle by checking if the sum of any two sides is greater than the third.

A graphic designer divided a 10 cm x 8 cm rectangular sheet into triangles to create a logo. The user is asked to find the length of the common side of triangles ΔAFD and ΔFED, calculate the semi-perimeter of ΔFED if DE = 11 cm, and find the area of ΔFED using the provided dimensions.

This problem involves calculating the common side length, semi-perimeter, and area of triangle ΔFED within a rectangle, using geometry concepts such as Heron's formula and triangle properties. Suitable for Grade 9 students.

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