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The diagram shows a sector \( OAB \) of a circle with center \( O \), radius \( 12 \, \text{cm} \), and a given angle \( AOB = 0.6 \, \text{radians} \). The point \( D \) is on \( OB \), and \( AD \) is perpendicular to \( OB \). The questions ask you to:

1. Find the lengths of \( AD \) and \( OD \).
2. Find the perimeter of the shaded region.
3. Calculate the area of the shaded region.

Let’s solve these one by one:

### (i) Lengths of \( AD \) and \( OD \):
To find these, we use trigonometric relationships in the right triangle \( OAD \).

- Using the cosine function:
  \[
  OD = 12 \cos(0.6) \, \text{cm}
  \]

- Using the sine function:
  \[
  AD = 12 \sin(0.6) \, \text{cm}
  \]

Let’s calculate these:
\[
OD \approx 12 \times 0.8253 = 9.9 \, \text{cm}
\]
\[
AD \approx 12 \times 0.5646 = 6.8 \, \text{cm}
\]

So, the lengths are:
- \( OD = 9.9 \, \text{cm} \)
- \( AD = 6.8 \, \text{cm} \)

### (ii) Perimeter of the shaded region:
The shaded region includes the arc \( AB \), the straight line segment \( AD \), and the line segment \( DB \).

1. The length of the arc \( AB \) is given by:
   \[
   \text{Arc length} = r \theta = 12 \times 0.6 = 7.2 \, \text{cm}
   \]

2. The length of \( AD \) is already found as \( 6.8 \, \text{cm} \).

3. The length of \( DB \) is the difference between \( OB \) and \( OD \):
   \[
   DB = OB - OD = 12 - 9.9 = 2.1 \, \text{cm}
   \]

The total perimeter is:
\[
\text{Perimeter} = \text{Arc length} + AD + DB = 7.2 + 6.8 + 2.1 = 16.1 \, \text{cm}
\]

### (iii) Area of the shaded region:
The shaded region is the sector area minus the area of triangle \( OAD \).

1. **Area of the sector** \( OAB \):
   \[
   \text{Sector area} = \frac{1}{2} r^2 \theta = \frac{1}{2} \times 12^2 \times 0.6 = 43.2 \, \text{cm}^2
   \]

2. **Area of triangle** \( OAD \):
   \[
   \text{Area of triangle} = \frac{1}{2} \times OD \times AD = \frac{1}{2} \times 9.9 \times 6.8 = 33.7 \, \text{cm}^2
   \]

The area of the shaded region is:
\[
\text{Shaded area} = \text{Sector area} - \text{Area of triangle} = 43.2 - 33.7 = 9.5 \, \text{cm}^2
\]

### Final Answers:
1. Lengths of \( AD = 6.8 \, \text{cm} \) and \( OD = 9.9 \, \text{cm} \).
2. Perimeter of the shaded region is \( 16.1 \, \text{cm} \).
3. Area of the shaded region is \( 9.5 \, \text{cm}^2 \).

Would you like further details or have any questions?

### 5 Related Questions:
1. How would the calculations change if the angle \( AOB \) were increased?
2. How do you calculate the length of an arc for any sector?
3. What is the difference between radian and degree measurements?
4. How would you calculate the area of a circle if only the sector angle is known?
5. How is the sine rule applied in triangles when the angles are not right-angled?

### Tip:
When working with radians, remember that \( \theta \) in radians corresponds to the arc length subtended by that angle on a unit circle. This makes many calculations simpler than working with degrees.

In the diagram, OAB is a sector of a circle with a radius of 12 cm and angle AOB = 0.6 radians. AD is perpendicular to OB. Find (i) the lengths of AD and OD, (ii) the perimeter of the shaded region, (iii) the area of the shaded region.

Learn how to calculate the lengths of AD and OD, the perimeter, and the area of a sector with a radius of 12 cm and a 0.6 radian angle. This problem covers key geometry concepts including the sine and cosine rules, arc length, and sector area. Suitable for students in Grades 10-12.

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