Question

Question 1
If Radii of two concentric circles are 4 cm and 5 cm, then length of each chord of one circle which is tangent to the other circle, is
(A) 3 cm
(B) 6 cm
(C) 9 cm
(D) 1 cm

Answer 1

Let O be the centre of two concentric circles $C_1$ and $C_2$, whose radii are $r_1=4cm$ and $r_2=5cm$.
Now, we draw a chord AC of circle $C_2$, which touches the circle $C_1$ at B.
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
$\therefore$ OB is perpendicular to AC.

Now in right-angled triangle, $\Delta OBC$, by using Pythagoras theorem.
$O{C}^2=B{C}^2=B{O}^2$
[$\because$ (hypotenuse)${}^2$ = (base)${}^2$ + ( perpendicular)${}^2$]
$\Rightarrow 5^2=B{C}^2+4^2\Rightarrow B{C}^2=25-16=9\Rightarrow BC=3cm$
$\therefore$ Length of chord AC = 2BC = 2$\times$ 3 = 6 cm
[$\because$ Line drawn from centre to the chord bisects the chord.]