Q3 of 63 Page 145

Two chords AB and CD of a circle whose lengths are 6 cm and 12 cm respectively, are parallel to each other and these lie in the same side of the centre of circle. If the distance between AB and CD be 3 cm, then find the radius of the circle.


Given chords AB = 6 cm and CD = 12 cm,


Draw OE⊥ AB intersecting CD at F,


As, AB||CD


OF⊥ CD [Corresponding angles]


Distance between AB and CD, EF= 3 cm


Now, we know that perpendicular from center to chord bisects the chord.


Then CF = FD = 1/2 CD = 6 cm


⇒ AE = EB = 1/2 AB = 3 cm


Let OF = y cm, OE = 3 + y cm and OD = OB = x cm = Radius.


Consider ΔOFD,


By Pythagoras Theorem,


⇒ OD2 = OF2 + FD2


⇒x2 = y2 + 62


⇒x2 = y2 + 36 … (1)


Consider ΔOEB,


By Pythagoras Theorem,


⇒ OB2 = OE2 + EB2


⇒x2 = (3 + y)2 + 32


⇒x2 = 9 + y2 + 6y + 9


⇒x2 = y2 + 6y + 18 … (2)


From (1) and (2),


⇒ y2 + 36 = y2 + 6y + 18


⇒ 36 = 6y + 18


⇒ 6y = 36 – 18


⇒ 6y = 18


⇒ y = 18/ 6


∴ y = 3


Substituting y value in (1),


⇒x2 = y2 + 36


⇒x2 = 32 + 36


⇒x2 = 9 + 36


⇒x2 = 45


[45 = 3 × 3 × 5]


∴ Radius = x = 3√5 cm


More from this chapter

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1

Write True/False in the following and give the reason of your answer if possible.

(i) AB and CD are chords of measure 3 cm and 4 cm respectively of a circle by which the angles subtended at the centre are respectively 70° and 50°.


(ii) Chords of a circle whose lengths are 10 cm and 8 cm are initiated at distances 8 cm and 5 cm respectively from the centre.


(iii) Out of the two chords AB and CD of a circle each is at a distance of 4 cm from the centre. Then AB = CD.


(iv) Congruent circles with centres O and O’ intersect at two points A and B. Then ∠AOB = ∠AO’B.


(v) A circle can be drawn through three collinear points.


(vi) A circle of radius 4 cm can be drawn through two points A and B of AB = 8 cm.

 2

If the radius of a circle is 13 cm and length of its one chord is 10 cm, then find the distance of this chord from the centre.

 4

In figure, two equal chords AB and CD intersect each other at E. Prove that arc DA = arc CB.

 5

In figure, AB and CD are equal chords of a circle. O is the centre of the circle of OM ⊥ AB and ON ⊥ CD. Then prove that ∠OMN = ∠ONM.