Advertisements
Advertisements
प्रश्न
If two chords AB and CD of a circle AYDZBWCX intersect at right angles (see figure), prove that arc CXA + arc DZB = arc AYD + arc BWC = semi-circle.
Advertisements
उत्तर
Given: In a circle AYDZBWCX, two chords AB and CD intersect at right angles.
To prove: arc CXA + arc DZB = arc AYD + arc BWC = Semi-circle.
Construction: Draw a diameter EF parallel to CD having centre M.
Proof: Since, CD || EF
arc EC = arc PD ...(i)
arc ECXA = arc EWB [Symmetrical about diameter of a circle]
arc AF = arc BF ...(ii)
We know that, ar ECXAYDF = Semi-circle
arc EA + arc AF = Semi-circle
⇒ arc EC + arc CXA = arc FB = Semi-circle ...[From equation (ii)]
⇒ arc DF + arc CXA + arc FB = Semi-circle ...[From equation (i)]
⇒ arc DF + arc FB + arc CXA = Semi-circle
⇒ arc DZB + arc C × A = Semi-circle
We know that, circle divides itself in two semi-circles, therefore the remaining portion of the circle is also equal to the semi-circle.
∴ arc AYD + arc BWC = Semi-circle
Hence proved.
APPEARS IN
संबंधित प्रश्न
Fill in the blanks:
A circle divides the plane, on which it lies, in __________ parts.
In the fig. a circle is inscribed in a quadrilateral ABCD in which ∠B = 90° if AD = 23cm,
AB = 29cm and DS = 5cm, find the radius of the circle.
A circle is inscribed in a ΔABC touching AB, BC and AC at P, Q and R respectively. If AB = 10 cm, AR = 7 cm and CR = 5 cm, find the length of BC.
In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the segments BD and DC into which BC is divided by the point of contact D, are of lengths 4cm and 3cm respectively. If the area of 2 ABC 21cm then find the lengths of sides AB and AC.
If ABCD is a cyclic quadrilateral in which AD || BC (In the given figure). Prove that ∠B = ∠C.
In the given figure, O is the centre of the circle and ∠BDC = 42°. The measure of ∠ACB is
In the given figure, the area enclosed between the two concentric circles is 770 cm2. If the radius of the outer circle is 21 cm, calculate the radius of the inner circle.
In the figure, O is the centre of a circle and diameter AB bisects the chord CD at a point E such that CE = ED = 8 cm and EB = 4 cm. The radius of the circle is
Say true or false:
Two diameters of a circle will necessarily intersect.
In the adjoining figure, AC is a diameter of the circle. AP = 3 cm and PB = 4 cm and QP ⊥ AB. If the area of ΔAPQ is 18 cm2, then the area of shaded portion QPBC is ______.
