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
Find the length of a tangent drawn to a circle with radius 5cm, from a point 13 cm from the center of the circle.
In the given figure, AB is a chord of length 16 cm of a circle of radius 10 cm. The tangents at A and B intersect at a point P. Find the length of PA.
Suppose You Are Given a Circle. Give a Construction to Find Its Centre.
In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB = 6 cm, BC = 9 cm and CD = 8 cm. Find the length of side AD.
If AB, BC and CD are equal chords of a circle with O as centre and AD diameter, than ∠AOB =
In the given figure, AB is a diameter of a circle with centre O and AT is a tangent. If \[\angle\] AOQ = 58º, find \[\angle\] ATQ.
In the given figure, PO \[\perp\] QO. The tangents to the circle at P and Q intersect at a point T. Prove that PQ and OTare right bisector of each other.
In the above figure, `square`XLMT is a rectangle. LM = 21 cm, XL = 10.5 cm. Diameter of the smaller semicircle is half the diameter of the larger semicircle. Find the area of non-shaded region.
In a right triangle ABC in which ∠B = 90°, a circle is drawn with AB as diameter intersecting the hypotenuse AC and P. Prove that the tangent to the circle at P bisects BC.
AB and AC are two equal chords of a circle. Prove that the bisector of the angle BAC passes through the centre of the circle.
