Question

state and prove cyclic quadrilateral?​

Answer 1

Answer:

The ratio between the diagonals and the sides can be defined and is known as Cyclic quadrilateral theorem. If there's a quadrilateral which is inscribed in a circle, then the product of the diagonals is equal to the sum of the product of its two pairs of opposite sides.

Step-by-step explanation:

Construct Cyclic Quadrilateral ABCD, where AB = 4cm, BC = 5 cm, AC = 6 cm and AD = 3cm.

Step I: Draw line AC = 6 cm

Step II: Draw two arcs of radius AB = 4 cm and BC = 5 cm with A and C as center respectively.

Step III: Join AB and BC to get the triangle ABC. Draw two perpendicular bisectors to any two sides of the triangle ABC.

Step IV: Draw a perpendicular bisector PQ to the side AC.

Step V: Draw a perpendicular bisector RS to the side AB.

Step VI: Mark the point of intersection of PQ and RS as ‘O’. Draw a circumcircle with point of intersection of two perpendicular bisectors as center and radius equal to distance from a vertices of triangle to point of intersection of perpendicular bisector.

Step VII: With ‘O’ as center and OA or OB or OC as radius, Draw the circumcircle of ΔABC. Draw an arc on the circle with the measurement equal to the length of the fourth side of the quadrilateral and join to form a cyclic quadrilateral.

Step VIII : With ‘A’ as center and radius as 3cm, draw an arc on the circle.

Step IX: Join AD and DC.

So, ABCD is the required Cyclic Quadrilateral.

Answer 2

Step-by-step explanation:

Cyclic quadrilateral states that a quadrilateral that is circumscribed in a circle is called a cyclic quadrilateral. It means that all the four vertices of the quadrilateral lie in the circumference of the circle.

Theorem 1:

In a cyclic quadrilateral, the sum of either pair of opposite angles is supplementary.

Proof: Let us now try to prove this theorem.

Given: A cyclic quadrilateral ABCD inscribed in a circle with center O.

Construction: Join the vertices A and C with center O.

(i) Consider arc ABC

∠AOC = 2∠ABC = 2α

The angle subtended by the same arc is the half of the angle subtended at the center.

(ii) Consider arc ADC

∠AOC = 2∠ADC = 2β

The angle subtended by the same arc is the half of the angle subtended at the center.

Consider both (i) and (ii)

∠AOC + Reflex ∠AOC = 360o

2∠ABC + 2∠ADC = 360o

2α + 2β = 360o

α + β = 180o

The converse of this theorem is also true, which states that if opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.

Theorem 2:

The ratio between the diagonals and the sides can be defined and is known as Cyclic quadrilateral theorem. If there’s a quadrilateral that is inscribed in a circle, then the product of the diagonals is equal to the sum of the product of its two pairs of opposite sides.

If PQRS is a cyclic quadrilateral, PQ and RS, and QR and PS are opposite sides. PR and QS are the diagonals.

(PQ x RS) + ( QR x PS) = PR x QS