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.

The correct option is √2.

Step-by-Step Explanation:

1. Analyze the given figure:

As shown in the image below:

We are given a semicircle with center O and diameter AB. Point C lies on the arc of the semicircle such that the line segment CO is perpendicular to AB (as indicated by the right-angle square symbol at point O).

2. Express lengths in terms of the radius (r):

Let the radius of the semicircle be r.

From the figure:

• The segment AO is a radius, so AO = r.

• The segment OB is a radius, so OB = r.

• The segment CO is also a radius, so CO = r.

• The diameter AB = AO + OB = r + r = 2r.

3. Calculate the lengths of chords AC and CB:

Consider the right-angled triangle ΔAOC (since CO ⊥ AB):

Using Pythagoras' theorem in ΔAOC:

${AC}^2={AO}^2+{CO}^2$

${AC}^2=r^2+r^2=2r^2$

$AC=\sqrt{2}r$

Similarly, for the right-angled triangle ΔBOC:

${CB}^2={OB}^2+{CO}^2$

${CB}^2=r^2+r^2=2r^2$

$CB=\sqrt{2}r$

4. Compute the required ratio:

We need to find the ratio:

$\frac{AC+CB}{AB}$

Substituting the values of AC, CB, and AB:

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

$=\frac{2\sqrt{2}r}{2r}$

$=\sqrt{2}$

Hence, the required ratio is √2.