The perimeter of a rhombus is $ 148 ext{ cm}$, and one of its diagonals is $ 24 ext{ cm}$. The area in $ ex

Question

The perimeter of a rhombus is $ 148	ext{ cm}$, and one of its diagonals is $ 24	ext{ cm}$. The area in $	ext{cm}^2 $ of the rhombus is:

Examoogle · Accepted Answer

The correct answer is Option B: 840. Calculate side ($a = 148/4 = 37$), use Pythagorean triplet with half-diagonal ($12$): $a^2 = 12^2 + (d_2/2)^2 \implies 37^2 = 12^2 + (d_2/2)^2 \implies 1369 - 144 = 1225 \implies d_2/2 = 35 \implies d_2 = 70$. Area $= 0.5 	imes 24 	imes 70 = 840$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Calculate side ($a = 148/4 = 37$), use Pythagorean triplet with half-diagonal ($12$): $a^2 = 12^2 + (d_2/2)^2 \implies 37^2 = 12^2 + (d_2/2)^2 \implies 1369 - 144 = 1225 \implies d_2/2 = 35 \implies d_2 = 70$. Area $= 0.5 	imes 24 	imes 70 = 840$.

Formula Used: $$Area = \frac{1}{2} 	imes d_1 	imes d_2$$, $$Side = \frac{Perimeter}{4}$$, $$Side^2 = (\frac{d_1}{2})^2 + (\frac{d_2}{2})^2$$

Given: Perimeter of the rhombus = 148 cm, Length of one diagonal ($d_1$) = 24 cm

Steps: Calculate the side of the rhombus: Since all sides of a rhombus are equal, divide the perimeter by 4: $a = 148 / 4 = 37$ cm.. Find the second diagonal: Use the property that diagonals bisect each other at right angles. Let $d_1 = 24$, so half-diagonal is 12. Using Pythagoras: $a^2 = (d_1/2)^2 + (d_2/2)^2 \implies 37^2 = 12^2 + (d_2/2)^2 \implies 1369 - 144 = 1225$. Thus, half-diagonal $d_2/2 = \sqrt{1225} = 35$, and $d_2 = 70$.. Calculate the Area: Use the area formula for a rhombus given two diagonals: $Area = 0.5 	imes d_1 	imes d_2 = 0.5 	imes 24 	imes 70 = 840$.

Answer: B is correct because the area calculated using the diagonals is 840 cm².

Exam Tip: This is a standard application of right-angled triangle properties within quadrilaterals, frequently seen in geometry sections of competitive exams.

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