Given a semicircle with O as the centre, as shown in the figure, the ratio $$\frac{{\overline {AC} + \overline {CB} }}{{\overline {AB} }}$$ is _____, where  $$\overline {AC} ,\;\overline {CB} \;\;and\;\;\overline {AB}$$ are chords.

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1. $$\sqrt 2$$
2. $$\sqrt 3$$
3. 2
4. 3

Option 1 : $$\sqrt 2$$
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Detailed Solution

Let the radius of the semicircle is r as shown in the diagram given below.

Now, OA = OC = OB = r

AB = OA + OB = 2r

$$\overline {AC} = \sqrt {{{\left( {OA} \right)}^2} + {{\left( {OC} \right)}^2}} = \sqrt {{r^2} + {r^2}} = \sqrt 2 r$$

$$\overline {BC} = \sqrt {{{\left( {OB} \right)}^2} + {{\left( {OC} \right)}^2}} = \sqrt {{r^2} + {r^2}} = \sqrt 2 r$$

The ratio $$\frac{{\overline {AC} + \overline {CB} }}{{\overline {AB} }} = \frac{{\sqrt 2 r + \sqrt 2 r}}{{2r}} = \sqrt 2$$
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