https://file.aiquickdraw.com/mathbot/file/1728359347046dw2b6cks.jpg

https://file.aiquickdraw.com/mathbot/file/1728359347046ieotmiln.jpg

Let's break down the parts of the question.

### Information given:
- The circle has a **radius of 6 cm**.
- The angle \(\angle AOB = 120^\circ\).
- We are asked to find:
  - **(f)**: Length of the minor arc \(AB\).
  - **(g)**: Length of the major arc \(AB\).
  - **(h)**: Perimeter of the shaded region.

We are also instructed to **use \(\pi = 3.14\)**.

### Formulas needed:
1. **Length of an arc** (for a central angle \(\theta\)) is given by:
   \[
   L = \frac{\theta}{360^\circ} \times 2\pi r
   \]
   where \(\theta\) is in degrees and \(r\) is the radius.

2. **Perimeter of the shaded region** involves both the length of the minor arc and the two radii \(OA\) and \(OB\).

### Step-by-step solutions:

#### (f) Length of the minor arc \(AB\):

- The minor arc corresponds to \(\angle AOB = 120^\circ\).
\[
L_{\text{minor}} = \frac{120^\circ}{360^\circ} \times 2\pi \times 6
= \frac{1}{3} \times 2 \times 3.14 \times 6
= 12.56 \, \text{cm}
\]

#### (g) Length of the major arc \(AB\):

- The major arc corresponds to the remaining angle of the circle, which is \(360^\circ - 120^\circ = 240^\circ\).
\[
L_{\text{major}} = \frac{240^\circ}{360^\circ} \times 2\pi \times 6
= \frac{2}{3} \times 2 \times 3.14 \times 6
= 25.12 \, \text{cm}
\]

#### (h) Perimeter of the shaded region:

The perimeter of the shaded region is the sum of the length of the minor arc and the two straight line segments \(OA\) and \(OB\) (each of which is the radius \(6 \, \text{cm}\)).

\[
P = L_{\text{minor}} + OA + OB
= 12.56 + 6 + 6
= 24.56 \, \text{cm}
\]

### Summary of results:
- **(f)**: Length of the minor arc \(AB = 12.56 \, \text{cm}\)
- **(g)**: Length of the major arc \(AB = 25.12 \, \text{cm}\)
- **(h)**: Perimeter of the shaded region = \(24.56 \, \text{cm}\)

Would you like more details on any part, or any further explanation?

---

### Related questions:
1. What is the area of the sector subtended by \(\angle AOB = 120^\circ\)?
2. How would the lengths of the arcs change if the radius was doubled?
3. What is the ratio of the length of the minor arc to the major arc?
4. How would the perimeter of the shaded region change if \(\angle AOB\) was halved?
5. What is the area of the triangle \(OAB\)?

### Tip:
Always check if a problem involving arcs or sectors provides you with the angle and radius. These two values will help solve nearly all aspects of such problems.

The diagram below shows a circle of radius 6 cm. AOB = 120°. [Use π = 3.14]. The problem asks to find: f) the length of the minor arc AB, g) the length of the major arc AB, h) the perimeter of the shaded region.

Learn how to calculate the length of the minor and major arcs AB and the perimeter of the shaded region for a circle with radius 6 cm and central angle 120°. Step-by-step solutions provided, suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation