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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Properties of Triangles

NTA Abhyas 2020 · Mathematics · Properties of Triangles

The lengths of two adjacent sides of a cyclic quadrilateral are 2...

The lengths of two adjacent sides of a cyclic quadrilateral are 22 units and 55 units and the angle between them is 60o{60}^{o}. If the area of the quadrilateral is 434\sqrt{3}& sq. units, then the perimeter of the quadrilateral isThe lengths of two adjacent sides of a cyclic quadrilateral are 2 units and 5 units and the angle between them is 60 ^

Options

  1. A12.512.5 units
  2. B1313 units
  3. C13.213.2 units
  4. D1212 units

Correct answer

D. 1212 units

Step-by-step solution

The lengths of two adjacent sides of a cyclic quadrilateral are 2 units and 5 units and the angle between them is 60 ^

cos60o=4+25−c22.2.5cos {60}^{o} = \dfrac{4 + 25 - c^{2}}{2.2. 5}

⇒10=29−c2⇒ 10 = 29 - c^{2}

⇒c2=19⇒ c^{2} = 19

⇒c=19⇒ c = \sqrt{19}

Now,

cos⁡120°=a2+b2−c22ab\cos 120° = \dfrac{a^{2} + b^{2} - c^{2}}{2ab}

⇒−12=a2+b2−192ab⇒- \dfrac{1}{2} = \dfrac{a^{2} + b^{2} - 19}{2ab}

⇒a2+b2−19=−ab⇒ a^{2} + b^{2} - 19 = - ab

⇒a2+b2+ab=19⇒ a^{2} + b^{2} + ab = 19

Area of quadrilateral =12×2×5sin⁡60°+12absin120°=43= \dfrac{1}{2} \times 2 \times 5\sin 60° + \dfrac{1}{2}absin 120° = 4\sqrt{3}

⇒532+ab34=43⇒ \dfrac{5\sqrt{3}}{2} + \dfrac{ab\sqrt{3}}{4} = 4\sqrt{3}

⇒ab4=4−52=32⇒ \dfrac{ab}{4} = 4 - \dfrac{5}{2} = \dfrac{3}{2}

⇒ab=6⇒ ab = 6

⇒a2+b2=13⇒ a^{2} + b^{2} = 13

⇒a=2,b=3⇒ a = 2, b = 3

Perimeter

=2+5+2+3= 2 + 5 + 2 + 3

=12= 12
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