The problem asks for the approximate length of the chord CD in a circle, given the lengths of other chords and segments.

Geometric Analysis of Triangle ABC

We are given points A, B, C on a circle with chord lengths AB = 5 cm, BC = 12 cm, and AC = 13 cm.

Let's check if $\triangle ABC$ is a right-angled triangle using the converse of the Pythagorean theorem:

• $AB^2 + BC^2 = 5^2 + 12^2 = 25 + 144 = 169$
• $AC^2 = 13^2 = 169$

Since $AB^2 + BC^2 = AC^2$, $\triangle ABC$ is a right-angled triangle with the right angle at B ($\angle ABC = 90^\circ$).

Circle Properties and Diameter

Because $\angle ABC$ is a right angle and A, B, C are points on the circle, the chord AC must be the diameter of the circle. Therefore, the diameter of the circle is 13 cm.

Geometric Analysis of Triangle ADC

A, D, C are also points on the circle. Since AC is the diameter, the angle subtended by the diameter at any point on the circumference is a right angle. Thus, $\angle ADC = 90^\circ$.

Now consider the right-angled triangle $\triangle ADC$:

• Hypotenuse AC = 13 cm
• One leg AD = 7 cm
• The other leg is CD, which we need to find.

Using the Pythagorean theorem in $\triangle ADC$:

$ AD^2 + CD^2 = AC^2 $

Substitute the known values:

$ 7^2 + CD^2 = 13^2 $

$ 49 + CD^2 = 169 $

Solve for $CD^2$:

$ CD^2 = 169 - 49 $

$ CD^2 = 120 $

Calculating and Approximating CD

Now, find the length of CD:

$ CD = \sqrt{120} $

To approximate $\sqrt{120}$:

• We know that $10^2 = 100$ and $11^2 = 121$.
• $\sqrt{120}$ is slightly less than $\sqrt{121}$, which is 11.
• $\sqrt{120} \approx 10.954$ cm.

The closest approximation among the given options is 11 cm.