Advertisements
Advertisements
प्रश्न
If bisectors of opposite angles of a cyclic quadrilateral ABCD intersect the circle, circumscribing it at the points P and Q, prove that PQ is a diameter of the circle.
Advertisements
उत्तर
Given: ABCD is a cyclic quadrilateral.
DP and QB are the bisectors of ∠D and ∠B, respectively.
To prove: PQ is the diameter of a circle.
Construction: Join QD and QC.
Proof: Since, ABCD is a cyclic quadrilateral.
∴ ∠CDA + ∠CBA = 180° ...[Sum of opposite angles of cyclic quadrilateral is 180°]
On dividing both sides by 2, we get
`1/2 ∠CDA + 1/2 ∠CBA = 1/2 xx 180^circ = 90^circ`
⇒ ∠1 + ∠2 = 90° ...(i) `[∠1 = 1/2 ∠CDA "and" ∠2 = 1/2 ∠CBA]`
But ∠2 = ∠3 [Angles in the same segment QC are equal] ...(ii)
∠1 + ∠3 = 90°
From equations (i) and (ii),
∠PDQ = 90°
Hence, PQ is a diameter of a circle, because diameter of the circle.
Subtends a right angle at the circumference.
APPEARS IN
संबंधित प्रश्न
A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.
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 figure m(arc LN) = 110°,
m(arc PQ) = 50° then complete the following activity to find ∠LMN.
∠ LMN = `1/2` [m(arc LN) - _______]
∴ ∠ LMN = `1/2` [_________ - 50°]
∴ ∠ LMN = `1/2` × _________
∴ ∠ LMN = __________
In the given figure, ∠BAD = 78°, ∠DCF = x° and ∠DEF = y°. Find the values of x and y.
ABCD is a cyclic quadrilateral in BC || AD, ∠ADC = 110° and ∠BAC = 50°. Find ∠DAC.
ABCD is a cyclic quadrilateral in ∠DBC = 80° and ∠BAC = 40°. Find ∠BCD.
Prove that the circles described on the four sides of a rhombus as diameters, pass through the point of intersection of its diagonals.
Circles are described on the sides of a triangle as diameters. Prove that the circles on any two sides intersect each other on the third side (or third side produced).
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.
Find all the angles of the given cyclic quadrilateral ABCD in the figure.
