The diagonals of a rhombus are in the ratio 3 : 4. Ifthe perimeter of the rhombus is 40 cm, then find itsarea

Sure, let's solve the problem step by step.

Step 1: Let's assume that the length of the shorter diagonal is 3x and the length of the longer diagonal is 4x, where x is a common factor.

Step 2: The perimeter of a rhombus is the sum of all its sides. Since a rhombus has four equal sides, each side will be 40 cm / 4 = 10 cm.

Step 3: In a rhombus, the diagonals bisect each other at right angles, dividing the rhombus into four congruent right-angled triangles.

Step 4: Using the Pythagorean theorem, we can find the length of each side of the right-angled triangle. Let's consider one of the triangles formed by the diagonals.

• The shorter diagonal (3x) is the hypotenuse of the right-angled triangle.
• The longer diagonal (4x) is the base of the right-angled triangle.
• The side of the rhombus (10 cm) is the perpendicular side of the right-angled triangle.

Applying the Pythagorean theorem, we have: (10/2)^2 + (4x/2)^2 = (3x)^2 5^2 + 2^2x^2 = 3^2x^2 25 + 4 = 9x^2 - 4x^2 29 = 5x^2 x^2 = 29/5 x = √(29/5)

Step 5: Now that we have the value of x, we can find the lengths of the diagonals:

• Length of the shorter diagonal = 3x = 3 * √(29/5)
• Length of the longer diagonal = 4x = 4 * √(29/5)

Step 6: The area of a rhombus can be calculated by multiplying the lengths of the diagonals and dividing by 2: Area = (3x * 4x) / 2 = (12x^2) / 2 = 6x^2

Step 7: Substitute the value of x to find the area: Area = 6 * (√(29/5))^2 = 6 * (29/5) = 34.8 cm^2

Therefore, the area of the rhombus is 34.8 cm^2.