The larger diagonal of a rhombus is 150% of its smaller diagonal, and its area is 432 $$cm^{2}$$. Find the length (in cm) of the side of the rhombus?

The larger diagonal of a rhombus is 150% of its smaller diagonal.

Let's assume the smaller diagonal is '2y'.

larger diagonal = 2y of 150%

= $$2y\times\frac{150}{100}$$

= 3y

its area is 432 $$cm^{2}$$.

area of rhombus = $$\frac{1}{2}\times\ product\ of\ both\ diagonals$$

$$432=\frac{1}{2}\times2y\times\ 3y$$

$$432=y\times\ 3y$$

y = 12

smaller diagonal = 2y = $$2\times12$$ = 24 cm

larger diagonal = 3y = $$3\times12$$ = 36 cm

As we know that, 4 $$\times$$ (length of each side of the rhombu)$$^2$$ = $$\left(larger\ diagonal\right)^2\ +\left(smaller\ diagonal\right)^2$$

4 $$\times$$ (length of each side of the rhombu)$$^2$$ = $$\left(36\right)^2\ +\left(24\right)^2$$

4 $$\times$$ (length of each side of the rhombu)$$^2$$ = $$\left(1296\right)\ +\left(576\right)$$

4 $$\times$$ (length of each side of the rhombu)$$^2$$ = 1872

(length of each side of the rhombu)$$^2$$ = 468

length of each side of the rhombu = $$\sqrt{468}$$

= $$6\sqrt{13}$$ cm