ABCDE is a pentagon. Ratio of sides AB : BC : CD : DE : EA ` is sqrt(2) : 1 : 2 : 1 : sqrt(2) `. If `angle A = 90^(@)` find its area if longer side is 6

Answer

To find the area of pentagon ABCDE with the given side ratios and angle, we can follow these steps:

### Step 1: Understand the Ratios
The sides of the pentagon are given in the ratio:
- AB : BC : CD : DE : EA = √2 : 1 : 2 : 1 : √2

Let’s denote the common unit for the ratios as \( x \). Therefore, we can express the lengths of the sides as:
- AB = \( \sqrt{2}x \)
- BC = \( 1x \)
- CD = \( 2x \)
- DE = \( 1x \)
- EA = \( \sqrt{2}x \)

### Step 2: Identify the Longer Side
We are given that the longer side of the pentagon is 6 cm. The longest side in the ratio is CD, which is \( 2x \). Therefore, we have:
\[ 2x = 6 \]
From this, we can solve for \( x \):
\[ x = \frac{6}{2} = 3 \]

### Step 3: Calculate the Lengths of Each Side
Now that we have \( x \), we can find the lengths of all the sides:
- AB = \( \sqrt{2} \times 3 = 3\sqrt{2} \) cm
- BC = \( 1 \times 3 = 3 \) cm
- CD = \( 2 \times 3 = 6 \) cm
- DE = \( 1 \times 3 = 3 \) cm
- EA = \( \sqrt{2} \times 3 = 3\sqrt{2} \) cm

### Step 4: Calculate the Area of Triangle ABE
Since angle A is 90 degrees, triangle ABE is a right triangle. The area of triangle ABE can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, both the base (AB) and height (EA) are equal to \( 3\sqrt{2} \):
\[ \text{Area}_{ABE} = \frac{1}{2} \times 3\sqrt{2} \times 3\sqrt{2} = \frac{1}{2} \times 9 \times 2 = 9 \text{ cm}^2 \]

### Step 5: Calculate the Area of Rectangle BCDE
Next, we can find the area of rectangle BCDE. The length is CD = 6 cm and the breadth is BC = 3 cm:
\[ \text{Area}_{BCDE} = \text{length} \times \text{breadth} = 6 \times 3 = 18 \text{ cm}^2 \]

### Step 6: Calculate the Total Area of Pentagon ABCDE
Finally, we can find the total area of the pentagon by adding the areas of triangle ABE and rectangle BCDE:
\[ \text{Total Area} = \text{Area}_{ABE} + \text{Area}_{BCDE} = 9 + 18 = 27 \text{ cm}^2 \]

### Final Answer
The area of pentagon ABCDE is \( 27 \text{ cm}^2 \).

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