Answer :
To complete Table 121 for the arcs of circles, we need to determine either the length of the arc or the angle at the center of the circle where it has been omitted. Here, we'll compute these values step by step.
### Step-by-Step Calculations
#### 1. Calculating the Length of the Arc:
The length of an arc (\( L \)) of a circle can be found using the formula:
[tex]\[ L = 2 \pi r \left(\frac{\theta}{360}\right) \][/tex]
where:
- \( r \) is the radius of the circle.
- \( \theta \) is the angle at the center of the circle.
For each given radius and angle at the center:
1) Radius \( 7 \, \text{cm} \), Angle \( 90^\circ \)
[tex]\[ L = 2 \pi \times 7 \times \frac{90}{360} = 10.995574287564276 \, \text{cm} \][/tex]
2) Radius \( 35 \, \text{m} \), Angle \( 72^\circ \)
[tex]\[ L = 2 \pi \times 35 \times \frac{72}{360} = 43.982297150257104 \, \text{m} \][/tex]
3) Radius \( 4.2 \, \text{cm} \), Angle \( 120^\circ \)
[tex]\[ L = 2 \pi \times 4.2 \times \frac{120}{360} = 8.79645943005142 \, \text{cm} \][/tex]
4) Radius \( 56 \, \text{m} \), Angle \( 135^\circ \)
[tex]\[ L = 2 \pi \times 56 \times \frac{135}{360} = 131.94689145077132 \, \text{m} \][/tex]
#### 2. Calculating the Angle at the Center:
To find the angle \( \theta \) at the center given the length of the arc \( L \) and radius \( r \), we rearrange the formula:
[tex]\[ \theta = \left(\frac{L}{2 \pi r}\right) \times 360 \][/tex]
For each given radius and length of the arc:
1) Radius \( 14 \, \text{m} \), Arc Length \( 11 \, \text{cm} \)
[tex]\[ \theta = \left(\frac{11}{2 \pi \times 14}\right) \times 360 = 45.01811247456468^\circ \][/tex]
2) Radius \( 21 \, \text{cm} \), Arc Length \( 22 \, \text{cm} \)
[tex]\[ \theta = \left(\frac{22}{2 \pi \times 21}\right) \times 360 = 60.02414996608624^\circ \][/tex]
### Completing the Table
Let’s fill in the table with these calculated values:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Radius} & \text{Angle at Centre} & \text{Length of Arc} \\ \hline 7 \, \text{cm} & 90^\circ & 10.995574287564276 \, \text{cm} \\ \hline 35 \, \text{m} & 72^\circ & 43.982297150257104 \, \text{m} \\ \hline 4.2 \, \text{cm} & 120^\circ & 8.79645943005142 \, \text{cm} \\ \hline 56 \, \text{m} & 135^\circ & 131.94689145077132 \, \text{m} \\ \hline 14 \, \text{m} & 45.01811247456468^\circ & 11 \, \text{cm} \\ \hline 21 \, \text{cm} & 60.02414996608624^\circ & 22 \, \text{cm} \\ \hline \end{array} \][/tex]
### Graphical Representation
For a graphical representation, you can sketch a circle for each row in the table, marking the radius and the angle. Draw the arc corresponding to the angle, and label it with the calculated length of the arc.
1. Radius \( 7 \, \text{cm} \), Angle \( 90^\circ \):
- Sketch a circle of radius \( 7 \, \text{cm} \).
- Draw a sector with a central angle of \( 90^\circ \).
- Label the arc length as \( 10.995574287564276 \, \text{cm} \).
2. Radius \( 35 \, \text{m} \), Angle \( 72^\circ \):
- Sketch a circle of radius \( 35 \, \text{m} \).
- Draw a sector with a central angle of \( 72^\circ \).
- Label the arc length as \( 43.982297150257104 \, \text{m} \).
3. Radius \( 4.2 \, \text{cm} \), Angle \( 120^\circ \):
- Sketch a circle of radius \( 4.2 \, \text{cm} \).
- Draw a sector with a central angle of \( 120^\circ \).
- Label the arc length as \( 8.79645943005142 \, \text{cm} \).
4. Radius \( 56 \, \text{m} \), Angle \( 135^\circ \):
- Sketch a circle of radius \( 56 \, \text{m} \).
- Draw a sector with a central angle of \( 135^\circ \).
- Label the arc length as \( 131.94689145077132 \, \text{m} \).
5. Radius \( 14 \, \text{m} \), Arc Length \( 11 \, \text{cm} \):
- Sketch a circle of radius \( 14 \, \text{m} \).
- Draw a sector with an arc length of \( 11 \, \text{cm} \).
- Label the central angle as \( 45.01811247456468^\circ \).
6. Radius \( 21 \, \text{cm} \), Arc Length \( 22 \, \text{cm} \):
- Sketch a circle of radius \( 21 \, \text{cm} \).
- Draw a sector with an arc length of \( 22 \, \text{cm} \).
- Label the central angle as \( 60.02414996608624^\circ \).
This completes the table and provides a clear, step-by-step solution for each calculation.
### Step-by-Step Calculations
#### 1. Calculating the Length of the Arc:
The length of an arc (\( L \)) of a circle can be found using the formula:
[tex]\[ L = 2 \pi r \left(\frac{\theta}{360}\right) \][/tex]
where:
- \( r \) is the radius of the circle.
- \( \theta \) is the angle at the center of the circle.
For each given radius and angle at the center:
1) Radius \( 7 \, \text{cm} \), Angle \( 90^\circ \)
[tex]\[ L = 2 \pi \times 7 \times \frac{90}{360} = 10.995574287564276 \, \text{cm} \][/tex]
2) Radius \( 35 \, \text{m} \), Angle \( 72^\circ \)
[tex]\[ L = 2 \pi \times 35 \times \frac{72}{360} = 43.982297150257104 \, \text{m} \][/tex]
3) Radius \( 4.2 \, \text{cm} \), Angle \( 120^\circ \)
[tex]\[ L = 2 \pi \times 4.2 \times \frac{120}{360} = 8.79645943005142 \, \text{cm} \][/tex]
4) Radius \( 56 \, \text{m} \), Angle \( 135^\circ \)
[tex]\[ L = 2 \pi \times 56 \times \frac{135}{360} = 131.94689145077132 \, \text{m} \][/tex]
#### 2. Calculating the Angle at the Center:
To find the angle \( \theta \) at the center given the length of the arc \( L \) and radius \( r \), we rearrange the formula:
[tex]\[ \theta = \left(\frac{L}{2 \pi r}\right) \times 360 \][/tex]
For each given radius and length of the arc:
1) Radius \( 14 \, \text{m} \), Arc Length \( 11 \, \text{cm} \)
[tex]\[ \theta = \left(\frac{11}{2 \pi \times 14}\right) \times 360 = 45.01811247456468^\circ \][/tex]
2) Radius \( 21 \, \text{cm} \), Arc Length \( 22 \, \text{cm} \)
[tex]\[ \theta = \left(\frac{22}{2 \pi \times 21}\right) \times 360 = 60.02414996608624^\circ \][/tex]
### Completing the Table
Let’s fill in the table with these calculated values:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Radius} & \text{Angle at Centre} & \text{Length of Arc} \\ \hline 7 \, \text{cm} & 90^\circ & 10.995574287564276 \, \text{cm} \\ \hline 35 \, \text{m} & 72^\circ & 43.982297150257104 \, \text{m} \\ \hline 4.2 \, \text{cm} & 120^\circ & 8.79645943005142 \, \text{cm} \\ \hline 56 \, \text{m} & 135^\circ & 131.94689145077132 \, \text{m} \\ \hline 14 \, \text{m} & 45.01811247456468^\circ & 11 \, \text{cm} \\ \hline 21 \, \text{cm} & 60.02414996608624^\circ & 22 \, \text{cm} \\ \hline \end{array} \][/tex]
### Graphical Representation
For a graphical representation, you can sketch a circle for each row in the table, marking the radius and the angle. Draw the arc corresponding to the angle, and label it with the calculated length of the arc.
1. Radius \( 7 \, \text{cm} \), Angle \( 90^\circ \):
- Sketch a circle of radius \( 7 \, \text{cm} \).
- Draw a sector with a central angle of \( 90^\circ \).
- Label the arc length as \( 10.995574287564276 \, \text{cm} \).
2. Radius \( 35 \, \text{m} \), Angle \( 72^\circ \):
- Sketch a circle of radius \( 35 \, \text{m} \).
- Draw a sector with a central angle of \( 72^\circ \).
- Label the arc length as \( 43.982297150257104 \, \text{m} \).
3. Radius \( 4.2 \, \text{cm} \), Angle \( 120^\circ \):
- Sketch a circle of radius \( 4.2 \, \text{cm} \).
- Draw a sector with a central angle of \( 120^\circ \).
- Label the arc length as \( 8.79645943005142 \, \text{cm} \).
4. Radius \( 56 \, \text{m} \), Angle \( 135^\circ \):
- Sketch a circle of radius \( 56 \, \text{m} \).
- Draw a sector with a central angle of \( 135^\circ \).
- Label the arc length as \( 131.94689145077132 \, \text{m} \).
5. Radius \( 14 \, \text{m} \), Arc Length \( 11 \, \text{cm} \):
- Sketch a circle of radius \( 14 \, \text{m} \).
- Draw a sector with an arc length of \( 11 \, \text{cm} \).
- Label the central angle as \( 45.01811247456468^\circ \).
6. Radius \( 21 \, \text{cm} \), Arc Length \( 22 \, \text{cm} \):
- Sketch a circle of radius \( 21 \, \text{cm} \).
- Draw a sector with an arc length of \( 22 \, \text{cm} \).
- Label the central angle as \( 60.02414996608624^\circ \).
This completes the table and provides a clear, step-by-step solution for each calculation.
