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Question
In the figure, a circle with center P touches the semicircle at points Q and C having center O. If diameter AB = 10, AC = 6, then find the radius x of the smaller circle.
Solution
Given: AB = 10 units, AC = 6 units, PC = PQ = x unit.
To find: x
Solution:
Diameter AB = 10 ...[Given]
∴ Radius = `1/2 xx "AB" = 1/2 xx 10` = 5.
∴ OQ = AO = OB = 5 ......(i)
AC = AO + OC .....[A–O–C]
∴ OC = AC – AO
∴ OC = 6 – 5 ......[Given and (i)]
∴ OC = 1 ......(ii)
OQ = OP + PQ ......[O–P–Q]
∴ OP = OQ – PQ
= 5 – x ......(iii) [From (i) and given]
Note that seg AB is tangent to the given smaller circle at point C.
∴ ∠PCO = 90° ......[Tangent theorem]
∴ In ∆PCO,
∠PCO = 90°
∴ OP2 = PC2 + OC2 ......[Pythagoras theorem]
∴ (5 – x)2 = x2 + (1)2 ......[From (ii) and (iii)]
∴ 25 – 10x + x2 = x2 + 1
∴ 10x = 24
∴ x = `24/10` = 2.4
∴ The radius x of the smaller circle is 2.4 units
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