Solution:
The chord of the larger circle is a tangent to the smaller circle as shown in the figure below.
PQ is a chord of a larger circle and a tangent of a smaller circle.
Tangent PQ is perpendicular to the radius at the point of contact S.
Therefore, ∠OSP = 90°
In ΔOSP (Right-angled triangle)
By the Pythagoras Theorem,
OP2 = OS2 + SP2
52 = 32 + SP2
SP2 = 25 - 9
SP2 = 16
SP = ± 4
SP is the length of the tangent and cannot be negative
Hence, SP = 4 cm.
QS = SP (Perpendicular from center bisects the chord considering QP to be the larger circle's chord)
Therefore, QS = SP = 4cm
Length of the chord PQ = QS + SP = 4 + 4
PQ = 8 cm
Therefore, the length of the chord of the larger circle is 8 cm.
☛ Check: NCERT Solutions Class 10 Maths Chapter 10
Video Solution:
Maths NCERT Solutions Class 10 Chapter 10 Exercise 10.2 Question 7
Summary:
If two concentric circles are of radii 5 cm and 3 cm, then the length of the chord of the larger circle which touches the smaller circle is 8 cm.
☛ Related Questions:
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Tangents PA and PB are drawn from an external point P to two concentric circles with centre O and radii 8 cm and 5 cm respectively, as shown in Fig.3. If AP =15 cm, then find the length of BP.
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Tangents PA and PB are drawn from an external point P to two concentric circles with centre O and radii 8 cm and 5 cm respectively, as shown in Fig. 3. If AP = 15 cm, then find the length of BP.
To find: BP
Construction: Join OP.
`Now ,OA _|_AP`and `OB_|_BP` `[therefore\text{Tangent to a circle is prependicular to the radius through the point of contact}]`
⇒ ∠OAP = ∠OBP = 90°
On applying Pythagoras theorem in ΔOAP, we obtain:
(OP)2 = (OA)2 + (AP)2
⇒ (OP)2 = (8)2 + (15)2
⇒ (OP)2 = 64 + 225
⇒ (OP)2 = 289
`rArr OP=sqrt289`
⇒ OP = 17
Thus, the length of OP is 17 cm.
On applying Pythagoras theorem in ΔOBP, we obtain:
(OP)2= (OB)2 + (BP)2
⇒ (17)2 = (5)2 + (BP)2
⇒ 289 = 25 + (BP)2
⇒ (BP)2 = 289 − 25
⇒ (BP)2 = 264
⇒ BP = 16.25 cm (approx.)
Hence, the length of BP is 16.25 cm.
Concept: Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
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