Explanation:

Let's break down the problem:

- Let the radius of the circle be r.

- Length (L) of rectangle = 10/7 × r

- Breadth (B) of rectangle = 5/14 × r

- Perimeter of rectangle = 2(L + B)

- Perimeter of circle = 2pr

- Sum of both perimeters = 2(L + B) + 2pr = 138

Substitute values:

- 2[(10/7)r + (5/14)r] + 2pr = 138

- Calculate inside bracket: (20/14)r + (5/14)r = (25/14)r

- 2 × (25/14)r = (50/14)r ˜ 3.571r

Full equation:

- 3.571r + 6.28r = 138 (since p ˜ 3.14)

- 9.851r = 138 ? r ˜ 14

Now, area of rectangle = L × B = (10/7)r × (5/14)r = 50/98 × r² = 25/49 × r²

Plug in r ˜ 14:

- r² = 196

- Area ˜ 25/49 × 196 = 100

Option 2 - 100 is correct.

Explanation of statements & options:

- The ratio links length and radius, so length is a multiple of radius.

- Breadth is also tied to radius.

- We're just solving for perimeter sums, substituting ratios, and calculating area.

- Other options don't fit after plugging values found.

So, 100 sq. cm, Option 2, is the correct answer!