From the image, I'll address question 17, which is the first visible problem in the uploaded document.

Problem 17: Given the arc $\angle APB = 120^\circ$ in a circle, where $AB$ is the chord and the radius of the circle is $6$ cm, you are asked to calculate the length of chord $AB$.

Solution: To find the length of chord $AB$, we can use the formula for the length of a chord given by: $AB = 2r \sin\left(\frac{\theta}{2}\right)$ where $r$ is the radius of the circle and $\theta$ is the angle subtended by the chord at the center of the circle in radians.

1. The radius $r$ is given as $6$ cm.
2. The angle $\theta$ subtended by the chord at the center is $120^\circ$.

First, convert $\theta$ into radians: $\theta_{\text{radians}} = \frac{120^\circ \times \pi}{180^\circ} = \frac{2\pi}{3} \text{ radians}$

Then, substitute into the chord length formula: $AB = 2 \times 6 \times \sin\left(\frac{2\pi}{6}\right) = 12 \times \sin\left(\frac{\pi}{3}\right)$ Since $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$: $AB = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3} \text{ cm}$

Thus, the length of the chord $AB$ is $6\sqrt{3} \text{ cm}$.

Would you like more details on this solution or help with another question? Here are some related questions you might consider:

1. How would the length of the chord change if the radius of the circle was different?
2. What would be the area of the segment formed between the chord $AB$ and the circle?
3. How does the length of the chord relate to the diameter of the circle in terms of the angle subtended at the center?
4. Can you solve for the circle's circumference given the radius used in this problem?
5. What would happen to the angle $\angle APB$ if the chord $AB$ was doubled in length?

Tip: Remember that for small angles, the approximation $\sin(\theta) \approx \theta$ (where $\theta$ is in radians) can provide quick estimates of lengths in circular motion problems.