The base of a triangle is equal to the perimeter of a square whose diagonal is $6 \sqrt{2}$ cm,and its height is equal to the side of a square whose area is 144 $cm^{2} $. The area of the triangle (in $cm^{2} $) is:

144

Diagonal of square = a\(\sqrt {2}\)

Area of right angled triangle = \(\frac{1}{2}\) × Base × Height

Perimeter of square = 4 × side 

We know,

a\(\sqrt {2}\)= 6\(\sqrt {2}\)

a = 6 side of square 

Perimeter = 4 × 6 = 24 = base of the triangle

Let the height = h

h² = 144

h = 12

Height of the triangle is equal to side of square

Area of triangle =\(\frac{1}{2}\) × 12× 24 = 144 cm2