Advertisements
Advertisements
प्रश्न
If a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are also equal.
Advertisements
उत्तर
Given: Let ABCD be a cyclic quadrilateral and AD = BC.
Join AC and BD.
To prove: AC = BD
Proof: In ΔAOD and ΔBOC,
∠OAD = ∠OBC and ∠ODA = ∠OCB ...[Since, same segments subtends equal angle to the circle]
AB = BC ...[Given]
ΔAOD = ΔBOC ...[By ASA congruence rule]
Adding is DOC on both sides, we get
ΔAOD + ΔDOC ≅ ΔBOC + ΔDOC
⇒ ΔADC ≅ ΔBCD
AC = BD ...[By CPCT]
APPEARS IN
संबंधित प्रश्न
If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.
Two chords AB and CD of lengths 5 cm 11cm respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6 cm, find the radius of the circle.
In the given figure, ∠BAD = 78°, ∠DCF = x° and ∠DEF = y°. Find the values of x and y.
In the given figure, ABCD is a cyclic quadrilateral. Find the value of x.
ABCD is a cyclic quadrilateral in ∠DBC = 80° and ∠BAC = 40°. Find ∠BCD.
If the two sides of a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are equal.
In the given figure, ABCD is a cyclic quadrilateral in which AC and BD are its diagonals. If ∠DBC = 55° and ∠BAC = 45°, find ∠BCD.
Prove that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent.
PQRS is a cyclic quadrilateral such that PR is a diameter of the circle. If ∠QPR = 67° and ∠SPR = 72°, then ∠QRS =
In the following figure, AOB is a diameter of the circle and C, D, E are any three points on the semi-circle. Find the value of ∠ACD + ∠BED.
