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The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60º.if the area of the quadrilatetal is 4root(3) ,then the perimeter of the quadrilateral is?
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The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the a...
- **Given information:**
- Two adjacent sides of the cyclic quadrilateral are 2 and 5.
- The angle between them is 60º.
- The area of the quadrilateral is 4√3.
- **Finding the diagonal:**
- Using the law of cosines, we can find the diagonal of the cyclic quadrilateral.
- Let the diagonal be 'd'.
- Using the formula: \( d^2 = a^2 + b^2 - 2ab \cos(\theta) \), where a = 2, b = 5, θ = 60º.
- \( d^2 = 2^2 + 5^2 - 2(2)(5)\cos(60º) \)
- \( d^2 = 4 + 25 - 20(\frac{1}{2}) \)
- \( d^2 = 4 + 25 - 10 \)
- \( d^2 = 19 \)
- \( d = \sqrt{19} \)
- **Calculating the perimeter:**
- The perimeter of the cyclic quadrilateral is the sum of all its sides.
- Given two adjacent sides are 2 and 5, and the diagonal is \( \sqrt{19} \).
- The other two sides can be found using the Pythagorean theorem, as the quadrilateral is cyclic.
- Let the other two sides be x and y.
- \( x^2 = 2^2 + (\sqrt{19})^2 \)
- \( x^2 = 4 + 19 = 23 \)
- \( x = \sqrt{23} \)
- Similarly, \( y^2 = 5^2 + (\sqrt{19})^2 \) gives \( y = \sqrt{44} = 2\sqrt{11} \)
- Therefore, the perimeter is: 2 + 5 + \( \sqrt{23} \) + \( 2\sqrt{11} \)
- **Final answer:**
- The perimeter of the cyclic quadrilateral is 7 + \( \sqrt{23} \) + \( 2\sqrt{11} \).
- Two adjacent sides of the cyclic quadrilateral are 2 and 5.
- The angle between them is 60º.
- The area of the quadrilateral is 4√3.
- **Finding the diagonal:**
- Using the law of cosines, we can find the diagonal of the cyclic quadrilateral.
- Let the diagonal be 'd'.
- Using the formula: \( d^2 = a^2 + b^2 - 2ab \cos(\theta) \), where a = 2, b = 5, θ = 60º.
- \( d^2 = 2^2 + 5^2 - 2(2)(5)\cos(60º) \)
- \( d^2 = 4 + 25 - 20(\frac{1}{2}) \)
- \( d^2 = 4 + 25 - 10 \)
- \( d^2 = 19 \)
- \( d = \sqrt{19} \)
- **Calculating the perimeter:**
- The perimeter of the cyclic quadrilateral is the sum of all its sides.
- Given two adjacent sides are 2 and 5, and the diagonal is \( \sqrt{19} \).
- The other two sides can be found using the Pythagorean theorem, as the quadrilateral is cyclic.
- Let the other two sides be x and y.
- \( x^2 = 2^2 + (\sqrt{19})^2 \)
- \( x^2 = 4 + 19 = 23 \)
- \( x = \sqrt{23} \)
- Similarly, \( y^2 = 5^2 + (\sqrt{19})^2 \) gives \( y = \sqrt{44} = 2\sqrt{11} \)
- Therefore, the perimeter is: 2 + 5 + \( \sqrt{23} \) + \( 2\sqrt{11} \)
- **Final answer:**
- The perimeter of the cyclic quadrilateral is 7 + \( \sqrt{23} \) + \( 2\sqrt{11} \).
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The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60º.if the area of the quadrilatetal is 4root(3) ,then the perimeter of the quadrilateral is?
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The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60º.if the area of the quadrilatetal is 4root(3) ,then the perimeter of the quadrilateral is? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60º.if the area of the quadrilatetal is 4root(3) ,then the perimeter of the quadrilateral is? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60º.if the area of the quadrilatetal is 4root(3) ,then the perimeter of the quadrilateral is?.
The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60º.if the area of the quadrilatetal is 4root(3) ,then the perimeter of the quadrilateral is? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60º.if the area of the quadrilatetal is 4root(3) ,then the perimeter of the quadrilateral is? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60º.if the area of the quadrilatetal is 4root(3) ,then the perimeter of the quadrilateral is?.
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