Question

AB and CD are two parallel chords of a circle with centre O such that AB = 6 cm and CD = 12 cm. The chords are on the same side of the centre and the distance between them is 3 cm. Construct line segment OL such that it bisects chords AB and CD perpendicularly at point L and M respectively.
Find the radius of the circle. (2 marks)

Answer 1

AB and CD are two parallel chords of a circle with centre O.

Let r be the radius of the circle.

AB = 6 cm, CD = 12 cm and LM = 3 cm

Join OC and OA.

Let OM = x, then OL = x + 3

In right $\Delta OCM$, $O{C}^2=O{M}^2+C{M}^2=(x{)}^2+{\left(\frac{CD}{2}\right)}^2$

$\Rightarrow r^2=x^2+{\left(\frac{12}{2}\right)}^2=x^2+36$ .......(i) (0.5 Marks)

In right $\Delta OAL$, $O{A}^2=O{L}^2+A{L}^2$

$\Rightarrow r^2=(x+3{)}^2+{\left(\frac{6}{2}\right)}^2=(x+3{)}^2+(3{)}^2$

$=(x+3{)}^2+9$ ............(ii) (0.5 Marks)

From (i) and (ii), $(x+3{)}^2+9=x^2+36$

$\Rightarrow x^2+6x+9+9=x^2+36$

$\Rightarrow 6x=36-18=18\Rightarrow x=\frac{18}{6}=3$

$\therefore (Radius{)}^2=r^2=x^2+36$

$=(3{)}^2+36=9+36=45$

$\therefore r=\sqrt{45}=3\sqrt{5}$ cm. (1 Mark)