Find the perimeter of a rhombus whose one diagonal is 16 cm long and area is 240 cm2 .

The correct option is 68 cm.

Let us break down the solution step-by-step to understand why this is the correct answer.

Step 1: Understand the formula for the area of a rhombus.
The area of a rhombus when its diagonals are known is given by the formula:

$$\text{Area}=\frac{1}{2}\times d_1\times d_2$$

where $d_1$ and $d_2$ are the lengths of the two diagonals.

Step 2: Find the length of the second diagonal.
We are given:
Area = 240 cm²
One diagonal ($d_1$) = 16 cm

Substitute these values into the area formula to find the other diagonal ($d_2$):

$$240=\frac{1}{2}\times 16\times d_2$$

Simplifying the equation:

$$240=8\times d_2$$

$$d_2=\frac{240}{8}=30\text{cm}$$

So, the length of the second diagonal is 30 cm.

Step 3: Relate the diagonals to the side length of the rhombus.
The diagonals of a rhombus bisect each other at right angles (90 degrees). Therefore, they divide the rhombus into four congruent right-angled triangles.
For any one of these right-angled triangles, the legs are half of the diagonals, and the hypotenuse is the side of the rhombus (let it be a).

The lengths of the half-diagonals are:

$$\frac{d_1}{2}=\frac{16}{2}=8\text{cm}$$

$$\frac{d_2}{2}=\frac{30}{2}=15\text{cm}$$

Applying the Pythagorean theorem to find the side length a:

$$a^2=8^2+{15}^2$$

$$a^2=64+225$$

$$a^2=289$$

Taking the square root on both sides:

$$a=\sqrt{289}=17\text{cm}$$

Thus, each side of the rhombus has a length of 17 cm.

Step 4: Calculate the perimeter.
Since all four sides of a rhombus are equal in length, the perimeter is given by:

$$\text{Perimeter}=4\times a$$

$$\text{Perimeter}=4\times 17=68\text{cm}$$

Therefore, the perimeter of the rhombus is 68 cm.