Step 1: Find the side and perimeter of the first square

The diagonal of the first square is given as $7\sqrt{2}$ cm. The diagonal of a square of side $s$ is $s\sqrt{2}$. Thus, the side $s_1 = 7$ cm. The perimeter of this square is $4 \times 7 = 28$ cm. Therefore, the base of the triangle is $28$ cm.

$\text{Base} = 4 \times 7 = 28 \text{ cm}$

Step 2: Find the side of the second square

The area of the second square is given as $169\text{ sq cm}$. The side $s_2$ of the square is the square root of its area: $s_2 = \sqrt{169} = 13$ cm. Therefore, the height of the triangle is $13$ cm.

$\text{Height} = \sqrt{169} = 13 \text{ cm}$

Step 3: Calculate the area of the triangle

Using the area formula for a triangle with base $b = 28$ cm and height $h = 13$ cm: $\text{Area} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 28 \times 13 = 14 \times 13 = 182\text{ sq cm}$.

$\text{Area} = \frac{1}{2} \times 28 \times 13 = 182 \text{ sq cm}$