The diagonals of a rhombus are in the ratio 3:4. If its perimeter is 40 cm, find the lengths of the sides and diagonals of the rhombus

To find the lengths of the sides and diagonals of the rhombus, we can start by assigning variables.

Let's say the shorter diagonal is 3x, and the longer diagonal is 4x.

Since a rhombus has two pairs of congruent sides, let's denote the length of a side as s.

The perimeter of the rhombus is the sum of all four sides, so we have:

P = s + s + s + s = 4s

Given that the perimeter is 40 cm, we can set up the equation:

4s = 40

Dividing both sides by 4, we find that s = 10.

Therefore, each side of the rhombus has a length of 10 cm.

To find the lengths of the diagonals, we can use the Pythagorean theorem.

In a rhombus, the diagonals bisect each other at right angles, creating four congruent right triangles. In each triangle, one leg is half the length of a side (s/2), and the other leg is the diagonal.

Using the shorter diagonal 3x and the side s, we can set up the equation:

(3x)^2 = (s/2)^2 + (s/2)^2

9x^2 = (s^2)/4 + (s^2)/4

9x^2 = (2s^2)/4

9x^2 = (2(10)^2)/4

9x^2 = 200/4

9x^2 = 50

Dividing both sides by 9, we find that x^2 ≈ 5.56

Taking the square root of both sides, we find that x ≈ 2.36

Therefore, the length of the shorter diagonal is approximately 3 * 2.36 = 7.08 cm.

Using the longer diagonal 4x, we can find its length:

4x = 4 * 2.36 = 9.42 cm.

So, the lengths of the sides and diagonals of the rhombus are as follows:

Sides: 10 cm
Shorter diagonal: 7.08 cm
Longer diagonal: 9.42 cm