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Let's work through this problem step by step. We are given:

- A circle with a radius of 2 cm, and center at \(O\).
- \(\angle XOY = 120^\circ\).
- We are asked to calculate:

  a. The length of arc \(XY\),  
  b. The area of sector \(XOY\),  
  c. The length of the chord \(XY\),  
  d. The length of the apothem to chord \(XY\).

---

### 1. **Arc Length \(XY\)**

The formula to find the length of an arc is:
\[
L = \theta \cdot \frac{r \cdot \pi}{180}
\]
Where:
- \(\theta = 120^\circ\) (central angle),
- \(r = 2\) cm (radius of the circle).

Substituting the values:
\[
L = 120 \cdot \frac{2 \cdot \pi}{180} = \frac{240 \pi}{180} = \frac{4\pi}{3} \, \text{cm}
\]
Thus, the length of the arc \(XY\) is \(\frac{4\pi}{3}\) cm or approximately **4.19 cm**.

---

### 2. **Area of Sector \(XOY\)**

The formula for the area of a sector is:
\[
A = \frac{\theta}{360} \cdot \pi r^2
\]
Substituting the values:
\[
A = \frac{120}{360} \cdot \pi \cdot 2^2 = \frac{1}{3} \cdot \pi \cdot 4 = \frac{4\pi}{3} \, \text{cm}^2
\]
Thus, the area of sector \(XOY\) is \(\frac{4\pi}{3}\) cm², or approximately **4.19 cm²**.

---

### 3. **Length of Chord \(XY\)**

The formula for the length of a chord is:
\[
\text{Chord length} = 2r \cdot \sin\left(\frac{\theta}{2}\right)
\]
Substituting the values:
\[
\text{Chord length} = 2 \cdot 2 \cdot \sin\left(\frac{120}{2}\right) = 4 \cdot \sin(60^\circ) = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3} \, \text{cm}
\]
Thus, the length of the chord \(XY\) is \(2\sqrt{3}\) cm, or approximately **3.46 cm**.

---

### 4. **Length of Apothem**

The apothem is the perpendicular distance from the center of the circle to the chord. We can calculate it using:
\[
\text{Apothem} = r \cdot \cos\left(\frac{\theta}{2}\right)
\]
Substituting the values:
\[
\text{Apothem} = 2 \cdot \cos(60^\circ) = 2 \cdot \frac{1}{2} = 1 \, \text{cm}
\]
Thus, the length of the apothem is **1 cm**.

---

Would you like further clarification or more details on any step?

### Additional Questions:
1. How does the formula for arc length change for different central angles?
2. What would happen if the radius was increased by 50%? How would the arc length and area change?
3. Can you derive the chord length formula for a given angle \(\theta\)?
4. How is the apothem useful in dividing the circle into equal parts?
5. What are the practical applications of calculating sector areas?

**Tip:** When solving problems involving circles, make sure to double-check if the angle is in degrees or radians, as this can affect your formulas.

Diketahui lingkaran berjari-jari 2 cm dengan pusat di titik O. Jika ∠XOY = 120°, hitunglah: a) panjang busur XY, b) luas juring XOY, c) panjang tali busur XY, d) panjang apotema ke tali busur XY.

This problem involves a circle with a radius of 2 cm and a central angle of 120°. Learn how to calculate the arc length XY, the area of sector XOY, the chord length XY, and the apothem using formulas for circles. Suitable for students in Grades 9-12.

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