Two concentric circles are of radius 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

# Two concentric circles are of radius 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

1. A
4 cm
2. B
8 cm
3. C
10 cm
4. D
12 cm

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### Solution:

Let ${C}_{1}$ be the circle with radius 3 cm and ${C}_{2}$ be the circle with radius 5 cm. Let O be the center of the circles.
Let the chord of ${C}_{2}$ that touches ${C}_{1}$ be BD. The point of contact as C.
OC = 3 cm,OB = OD = 5 cm.
Tangent at any point of a circle is perpendicular to the radius through the point of contact.
OCBD
In triangle OBC,
${\mathit{OB}}^{2}={\mathit{OC}}^{2}+{\mathit{BC}}^{2}$
$⇒{5}^{2}={3}^{2}+{\mathit{BC}}^{2}$
$⇒{\mathit{BC}}^{2}=25-9=16$

Also, CD = 4 cm
BD = BC + CD = 8 cm
So, Option 2 is correct.

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