
The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is . If the area of the quadrilateral is , 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 . So the angle opposite to will be
We draw diagram of the given cyclic quadrilateral where we assign the angles and . We assign the given lengths and unknown lengths of four t sides to .
We know from the sine rule that area of a triangle with lengths with two sides and the angle made by them is given by ,
Putting in equation(1) we get area of the triangle
Similarly the area of the triangle is ,
It is also given that the area of the quadrilateral is but the area of quadrilateral is the sum of area of the triangle and area of the triangle . So,
We also know from the cosine rule that the relation in the triangle among three sides and the angle made by is given by ,
Similarly in the triangle
So we equate right hand side of equation (2) and (3) and put known values
We obtained that . So we form the quadratic equation in whose roots are and solve it by splitting the middle term method
Now or . We reject the negative values for distances to get .
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.
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

We draw diagram of the given cyclic quadrilateral
We know from the sine rule that area of a triangle with lengths with two sides
Putting in equation(1)
Similarly the area of the triangle
It is also given that the area of the quadrilateral
We also know from the cosine rule that the relation in the triangle
Similarly in the triangle
So we equate right hand side of equation (2) and (3) and put known values
We obtained that
Now
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.
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