The perimeter of a rhombus is $100 ext{ cm}$ and one of the diagonals is $40 ext{ cm}$. Then the area of t

Question

The perimeter of a rhombus is $100 	ext{ cm}$ and one of the diagonals is $40 	ext{ cm}$. Then the area of the rhombus is

Examoogle · Accepted Answer

The correct answer is Option A: $1200 	ext{ cm}^2$. Use the property that diagonals of a rhombus bisect each other at 90 degrees. Side $a = 100/4 = 25$. Half-diagonal $d_1/2 = 20$. The other half-diagonal $d_2/2 = \sqrt{25^2 - 20^2} = \sqrt{625 - 400} = 15$. So $d_2 = 30$. Area = $0.5 	imes 40 	imes 30 = 600$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Use the property that diagonals of a rhombus bisect each other at 90 degrees. Side $a = 100/4 = 25$. Half-diagonal $d_1/2 = 20$. The other half-diagonal $d_2/2 = \sqrt{25^2 - 20^2} = \sqrt{625 - 400} = 15$. So $d_2 = 30$. Area = $0.5 	imes 40 	imes 30 = 600$.

Formula Used: $$Area = \frac{1}{2} 	imes d_1 	imes d_2$$

Given: Perimeter of the rhombus = 100 cm, length of one diagonal (d1) = 40 cm.

Steps: Calculate Side Length: Since all sides of a rhombus are equal, the side $a$ is given by perimeter divided by 4: $a = 100/4$.. Find the Second Diagonal: Diagonals of a rhombus bisect each other at right angles. Using Pythagoras theorem in the triangle formed by the side and two half-diagonals: $(d_1/2)^2 + (d_2/2)^2 = a^2$. Given $d_1 = 40$, so $d_1/2 = 20$.. Calculate Area: Using the area formula for a rhombus involving diagonals:

Answer: B is correct because the second diagonal is 30 cm, resulting in an area of 600 cm².

Exam Tip: This property of right-angled bisecting diagonals is frequently used in coordinate geometry to find vertices or area of rhombi plotted on a Cartesian plane.

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