Question:
The perimeter of a rhombus is 100 cm and one of its diagonals is 40 cm. Find the area of the rhombus.
एक समचतुर्भुज का परिमाप 100 सेमी है और इसका एक विकर्ण 40 सेमी है। समचतुर्भुज का क्षेत्रफल ज्ञात कीजिए।
Options:
- 600 cm\(^2\)
- 500 cm\(^2\)
- 480 cm\(^2\)
- 720 cm\(^2\)
- 600 सेमी\(^2\)
- 500 सेमी\(^2\)
- 480 सेमी\(^2\)
- 720 सेमी\(^2\)
Correct Answer: a)
Explanation:
Perimeter of rhombus = \(4a = 100 \implies a = 25\) cm.
Diagonals of a rhombus bisect each other at right angles.
Let diagonals be \(d_1 = 40\) cm and \(d_2\).
Semi-diagonal \(\frac{d_1}{2} = 20\) cm.
In the right-angled triangle formed by semi-diagonals and side:
\(\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = a^2\)
\(20^2 + \left(\frac{d_2}{2}\right)^2 = 25^2 \implies 400 + \left(\frac{d_2}{2}\right)^2 = 625\)
\(\left(\frac{d_2}{2}\right)^2 = 225 \implies \frac{d_2}{2} = 15 \implies d_2 = 30\) cm.
Area of rhombus = \(\frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 40 \times 30 = 600\) cm\(^2\).
Diagonals of a rhombus bisect each other at right angles.
Let diagonals be \(d_1 = 40\) cm and \(d_2\).
Semi-diagonal \(\frac{d_1}{2} = 20\) cm.
In the right-angled triangle formed by semi-diagonals and side:
\(\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = a^2\)
\(20^2 + \left(\frac{d_2}{2}\right)^2 = 25^2 \implies 400 + \left(\frac{d_2}{2}\right)^2 = 625\)
\(\left(\frac{d_2}{2}\right)^2 = 225 \implies \frac{d_2}{2} = 15 \implies d_2 = 30\) cm.
Area of rhombus = \(\frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 40 \times 30 = 600\) cm\(^2\).
भुजा \(a = \frac{100}{4} = 25\) सेमी।
\(d_1 = 40\) सेमी \(\implies d_1/2 = 20\) सेमी।
पाइथागोरस प्रमेय से: \((d_2/2)^2 = 25^2 – 20^2 = 625 – 400 = 225 \implies d_2/2 = 15 \implies d_2 = 30\) सेमी।
क्षेत्रफल = \(\frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 40 \times 30 = 600\) सेमी\(^2\)।
\(d_1 = 40\) सेमी \(\implies d_1/2 = 20\) सेमी।
पाइथागोरस प्रमेय से: \((d_2/2)^2 = 25^2 – 20^2 = 625 – 400 = 225 \implies d_2/2 = 15 \implies d_2 = 30\) सेमी।
क्षेत्रफल = \(\frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 40 \times 30 = 600\) सेमी\(^2\)।