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The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60. If the area of the quadrilateral is 43, find the remaining two sides.

Answer
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Hint: Draw a diagram of the given cyclic quadrilateral and assign all the given data in the question. Use the law of sine and cosine rule to find out the length of other sides.

Complete step by step answer:
We know that the cyclic quadrilateral is inscribed within a circle or the vertices of a cyclic quadrilateral lies on a circle. We also know that the sum of opposite angles in a quadrilateral is 180. So the angle opposite to 60 will be 18060=120
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We draw diagram of the given cyclic quadrilateral ABCD where we assign the angles ABC=60 and ADC=60. We assign the given lengths and unknown lengths of four t sides to AB=5,BC=2,CD=c,DA=d.
We know from the sine rule that area of a triangle with lengths with two sides a,b and the angle made by them ABC is given by ,
Δ=12absin(ABC)...(1)
Putting in equation(1) b=AB=5,a=BC=2,ABC=60we get area of the triangle ABC
Δ1=12absin(ABC)=12(5232)=532
Similarly the area of the triangle ADC is ,
Δ1=12cdsin(ADC)=12(cd32)=cd34
It is also given that the area of the quadrilateral ABCD is Δ=43 but the area of quadrilateral ABCD is the sum of area of the triangle ABC and area of the triangle ADC. So,
Δ=Δ1+Δ243=532+cd34cd=1610=6

We also know from the cosine rule that the relation in the triangle ABC among three sides AB,BC,AC and the angle made by AB,BC=ABC is given by ,
AC2=AB2+BC22ABBCcos(ABC)...(2)
Similarly in the triangle ADC
AC2=AD2+CD22ADCDcos(ADC)...(3)
So we equate right hand side of equation (2) and (3) and put known values
AB2+BC22ABBCcos(ABC)=AD2+CD22ADCDcos(ADC)22+522(2)(5)cos(60)=d2+c22cdcos(120)19=d2+c2+cdd2+c2=13(cd=6)
We obtained that d2+c2=13,c2d2=36. So we form the quadratic equation in x whose roots are c2,d2 and solve it by splitting the middle term method
x2(c2+d2)x+cd=0x213x+36=0(x9)(x4)=0x=9 or x=4
Now c2=9,d2=4 or c2=4,d2=9. We reject the negative values for distances to get c=3,d=2 or c=2,d=3.

So the length of other two sides are 2 and 3.

Note: The question tests your knowledge of cyclic quadrilateral, sine rule and cosine rule. Careful solving of simultaneous equations with substitution and usage of formula will lead us to arrive at the correct result. The question can also be reframed to find the rest of the angles.