Question

Quadrilateral PQRS is cyclic quadrilateral if measure angle p=60° then find measures angle r=?

Answer 1

Cyclic quadrilateral :

• In a cyclic quadrilateral, the opposite angles add up to 180°.
• Exterior angle is equal to the opposite interior angle.
• All the four vertices of a quadrilateral inscribed in a circle lie on the circumference of the circle.

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In cyclic Quadrilateral PQRS:

if Angle p = 60°

Angle r = 180° - 60°

Angle r = 120° (opposite angles are supplementary)

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Extras :

Radius of the cyclic quadrilateral:

Assuming that a,b, c and d are the successive sides of the cyclic quadrilateral and 'x' being it's semi perimeter,

Radius :

$\dfrac{1}{4}\sqrt{ \dfrac{(ab + cd)(ac + bd)(ad + bc) }{(x - a)(x - b)(x - c)(x - d)} }$

Diagonals of cyclic quadrilateral:

Assuming a,b,c and d to be the sides of the cyclic quadrilateral and d and y to be the diagonals,

$d \: = \sqrt{ \dfrac{(ac + bd)(ad + bc)}{ab + cd} }$

and

$y = \sqrt{ \dfrac{(ac + bd)(ab + cd)}{ad + bc} }$

Area of the cyclic quadrilateral:

Assuming a,b,c and d to be the sides of the cyclic quadrilateral, the area is:

$s = \dfrac{ a + b + c }{2}$

s being the semi perimeter.

Area :

$\sqrt{(s - a)(s - b)(s - c)(s - d)}$

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