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.

Answer 1

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