https://file.aiquickdraw.com/mathbot/file/1727800374544i88t432t.jpg

To calculate the area of the minor sector \( OAC \), we need to break the problem down step by step. The key information provided in the image includes:

- The radius \( OC = OA = 19 \, \text{mm} \) (as tangents to a circle meet the radius at 90°).
- The angle \( \angle AOB = 34^\circ \).

### Step 1: Determine the angle for sector \( OAC \)
Since \( AB \) and \( BC \) are tangents to the circle, and \( \angle AOB = 34^\circ \), the angle for the minor sector \( OAC \) can be calculated as:
\[
\text{Angle of sector } OAC = 360^\circ - 2 \times 34^\circ = 292^\circ
\]
Thus, the central angle for the sector is \( 68^\circ \).

### Step 2: Area formula for a sector
The area \( A \) of a sector is given by the formula:
\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\]
Where:
- \( \theta = 68^\circ \) is the angle of the sector.
- \( r = 19 \, \text{mm} \) is the radius of the circle.

### Step 3: Calculate the area
Substituting the values into the formula:
\[
A = \frac{68^\circ}{360^\circ} \times \pi \times (19^2)
\]
\[
A = \frac{68}{360} \times \pi \times 361
\]
\[
A \approx 0.1889 \times \pi \times 361
\]
\[
A \approx 0.1889 \times 1134.11
\]
\[
A \approx 214.15 \, \text{mm}^2
\]

Thus, the area of the minor sector \( OAC \) is approximately **214.2 mm²** (rounded to 1 decimal place).

### Reasoning:
- The angle at the center for the minor sector was calculated by subtracting the known angles from 360°.
- The area of a sector formula was used based on the radius and the angle provided.

Would you like further details on the steps, or do you have any questions?

Here are 5 related questions:
1. How would the area change if the angle \( AOB \) was increased?
2. What is the area of the major sector in this circle?
3. How would you calculate the arc length of sector \( OAC \)?
4. How does the radius affect the area of a sector in general?
5. Can you derive the formula for the area of a sector?

**Tip:** Always double-check the given angles and whether you're working with the major or minor sector when calculating areas.

Calculate the area of the minor sector OAC, to 1 d.p. Give reasons for your answer.

Learn how to calculate the area of the minor sector OAC in a circle with a radius of 19 mm and an angle of 68°. The solution includes key formulas, such as the sector area formula, and relevant theorems like the tangent to circle properties. Suitable for Grades 9-11.

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