Q13 of 35 Page 11

In the given figure, a circle touches all the four sides of a quadrilateral ABCD whose three sides are AB = 6cm, BC = 7cm, and CD = 4cm. Find AD.

Mark the touching points as P, Q, R and S as shown



As tangents from a point are of equal length we have


AQ = AR = a


BR = BS = b


CP = CS = c


DP = DQ = d


From figure


⇒ BC = BS + SC


⇒ 7 = b + c


⇒ b = 7 – c … BC is 7 cm given …(i)


Also,


⇒ DC = DP + PC


⇒ 4 = d + c


⇒ c = 4 – d … DC is 4 cm given …(ii)


And


⇒ AB = AR + RB


⇒ 6 = a + b … AB is 6 cm given


⇒ a = 6 – b


Using (i)


⇒ a = 6 – (7 – c)


Using (ii)


⇒ a = 6 – (7 – (4 – d))


⇒ a = 6 – (7 – 4 + d)


⇒ a = 6 – 7 + 4 – d


⇒ a + d = 3


⇒ AQ + QD = 3 …since AQ = a and QD = d


From figure AQ + QD = AD


⇒ AD = 3 cm


Hence AD is 3 cm


More from this chapter

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11

In the given figure, O is the centre of two concentric circles of radii 4 cm and 6cm respectively. PA and PB are tangents to the outer and inner circle respectively. If PA=10cm, find the length of PB up to one place of decimal.

 12

Show that the tangents at the extremities of any chord of a circle make equal angles with the chord.

 14

(i) From an external point P, tangents PA and PB are drawn to a circle with centre O. If CD is the tangent to the circle at the point E and PA = 14 cm, find the perimeter of PCD.

(ii) If PA = 11cm, PD = 7 cm, then DE = ?


 15

In two concentric circles, prove that all chords of the outer circle which touch the inner arc of equal length.