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The problem involves two similar pentagons, \( OABCD \) and \( OWXYZ \), where the shaded area between these pentagons is given as \( 102 \, \text{cm}^2 \). The lengths of two sides are provided:
- \( CD = 6 \, \text{cm} \)
- \( YZ = 11 \, \text{cm} \).

Since the pentagons are similar, the ratio of the sides of the two pentagons can help us find the relationship between their areas. Let's outline the steps to find the area of pentagon \( OABCD \).

### Step 1: Ratio of the sides
The ratio of the lengths of corresponding sides of similar shapes gives us the linear scale factor, \( k \). Here, the corresponding sides are \( CD \) and \( YZ \):
\[
k = \frac{YZ}{CD} = \frac{11}{6}
\]

### Step 2: Ratio of the areas
The ratio of the areas of two similar figures is the square of the ratio of their corresponding sides. So, the ratio of the areas of the pentagons \( OWXYZ \) and \( OABCD \) is:
\[
\text{Area ratio} = k^2 = \left(\frac{11}{6}\right)^2 = \frac{121}{36}
\]

Let \( A_{OABCD} \) be the area of the smaller pentagon \( OABCD \), and \( A_{OWXYZ} \) the area of the larger pentagon \( OWXYZ \). Using the area ratio:
\[
\frac{A_{OWXYZ}}{A_{OABCD}} = \frac{121}{36}
\]
\[
A_{OWXYZ} = \frac{121}{36} A_{OABCD}
\]

### Step 3: Shaded area equation
The shaded area is the difference between the area of the larger pentagon \( OWXYZ \) and the smaller pentagon \( OABCD \):
\[
A_{\text{shaded}} = A_{OWXYZ} - A_{OABCD}
\]
Substituting the relationship from Step 2:
\[
102 = \frac{121}{36} A_{OABCD} - A_{OABCD}
\]
Factoring out \( A_{OABCD} \):
\[
102 = A_{OABCD} \left( \frac{121}{36} - 1 \right)
\]
Simplify:
\[
102 = A_{OABCD} \left( \frac{121}{36} - \frac{36}{36} \right)
\]
\[
102 = A_{OABCD} \left( \frac{85}{36} \right)
\]
Now solve for \( A_{OABCD} \):
\[
A_{OABCD} = \frac{102 \times 36}{85} = \frac{3672}{85} \approx 43.2 \, \text{cm}^2
\]

### Final Answer:
The area of pentagon \( OABCD \) is approximately \( 43.2 \, \text{cm}^2 \).

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:
1. How do the properties of similar polygons affect their areas?
2. What would happen to the area ratio if the linear scale factor was doubled?
3. How is the shaded area affected if the side lengths of both pentagons are doubled?
4. Can you use a similar method to find areas of other polygons (e.g., hexagons)?
5. How would the problem change if the larger pentagon's area was given directly?

**Tip**: In problems involving similar shapes, remember that the ratio of areas is the square of the ratio of corresponding sides.

The diagram shows two similar pentagons OABCD and OWXYZ. The length of CD is 6 cm and the length of YZ is 11 cm. The shaded area is 102 cm². Find the area of pentagon OABCD.

Solve a geometry problem involving two similar pentagons OABCD and OWXYZ. The shaded area is 102 cm², and the side lengths CD and YZ are given as 6 cm and 11 cm respectively. Learn how to find the area of pentagon OABCD by applying ratios of areas.

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