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 point, whose distance from the centre of a circle is greater than its radius lies in __________ of the circle. (exterior/ interior)
In fig.. O is the center of the circle and BCD is tangent to it at C. Prove that ∠BAC +
∠ACD = 90°
Fill in the blank
Circles having the same centre and different radii are called ...........................circles.
In the given figure, PA and PB are the tangent segments to a circle with centre O. Show that the points A, O, B and P are concyclic.
In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm such that the segments BD and DC into which BC is divided by the point of contact D, are of lengths 6 cm and 9 cm respectively. If the area of ΔABC = 54 cm2 then find the lengths of sides AB and AC.
In Fig. 5, the chord AB of the larger of the two concentric circles, with centre O, touches the smaller circle at C. Prove that AC = CB.
One chord of a circle is known to be 10 cm. The radius of this circle must be
Use the figure given below to fill in the blank:
EF is a ______ of the circle.
All the radii of a circle are _______________
A point A is 26 cm away from the centre of a circle and the length of the tangent drawn from A to the circle is 24 cm. Find the radius of the circle.
