The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter :

Question

Accepted Answer

To find the perimeter of a rhombus when the lengths of its diagonals are given, we can use the properties of a rhombus and the Pythagorean theorem. A rhombus is a type of quadrilateral where all sides are equal in length, and its diagonals bisect each other at right angles.

Given:
- Diagonal 1 = 16 cm
- Diagonal 2 = 30 cm

Steps to find the perimeter:

1. **Calculate the length of one side of the rhombus:**
   - The diagonals of a rhombus bisect each other at right angles, forming four right-angled triangles within the rhombus.
   - Each side of the rhombus is the hypotenuse of one of these right-angled triangles.
   - The legs of the triangle are half of each diagonal:
     - Half of diagonal 1 = 16 cm / 2 = 8 cm
     - Half of diagonal 2 = 30 cm / 2 = 15 cm

2. **Use the Pythagorean theorem to find the length of one side:**
   $$
   \text{Side} = \sqrt{(8 \, \text{cm})^2 + (15 \, \text{cm})^2}
   $$
   $$
   \text{Side} = \sqrt{64 + 225}
   $$
   $$
   \text{Side} = \sqrt{289}
   $$
   $$
   \text{Side} = 17 \, \text{cm}
   $$

3. **Calculate the perimeter of the rhombus:**
   - Since all sides of a rhombus are equal, the perimeter is four times the length of one side.
   $$
   \text{Perimeter} = 4 \times 17 \, \text{cm} = 68 \, \text{cm}
   $$

Therefore, the perimeter of the rhombus is 68 cm, which corresponds to Option 3.

**Explanation of why other options are incorrect:**

- **Option 1: 63.5 cm** - This value is incorrect because it does not match the calculated perimeter of 68 cm using the Pythagorean theorem and the properties of a rhombus.
  
- **Option 2: 65 cm** - This value is also incorrect for the same reason; it does not align with the calculated perimeter of 68 cm.

- **Option 4: 70.2 cm** - This value exceeds the correct perimeter calculation and does not fit the mathematical derivation based on the given diagonal lengths.

In conclusion, the correct perimeter of the rhombus, based on the given diagonal measurements, is 68 cm, which is why Option 3 is the correct answer.