A rectangle of side 8cm\times 6cm and a rhombus
A rectangle of side 8cm × 6cm and a rhombus (side = 8cm, acute angle $ 60^\circ$ are joined along one 8 cm side. what’s the ratio of their areas (rectangle : rhombus)?
8 सेमी × 6 सेमी भुजा वाला एक आयत और एक समबाहु चतुर्भुज जिसकी भुजा = 8 सेमी और तीव्र कोण $60^\circ$ है,एक 8 सेमी की भुजा के अनुदिश जुड़े हुए है। उनके क्षेत्रफलों का अनुपात (आयत : समचतुर्भुज) ?
Detailed Solution & Logic
$ \sqrt{3} : 2 $
Area of rectangle:
$8 \times 6 = 48$
Area of rhombus:
$8^2 \times \sin 60^\circ = 64 \times \frac{\sqrt{3}}{2} = 32\sqrt{3}$
Now, ratio (rectangle : rhombus):
$48 : 32\sqrt{3}$
Simplify:
Divide by 16 → $3 : 2\sqrt{3}$
Rationalize:
$= \frac{3}{2\sqrt{3}} = \frac{3}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2}$
So,
$= \sqrt{3} : 2$
Answer: $\sqrt{3} : 2$
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