Answer :
Answer:
Let's break down each of your requests step by step.
### 1. Finding the number of sides of a regular polygon given the measure of its exterior angles in the ratio 2:3:4:9
To clarify, let's assume you are asking for the number of sides of a regular polygon where the exterior angle ratio is 2:3:4:9. This might be interpreted as the ratios of the measures of exterior angles of four different polygons.
However, if you are asking about a single regular polygon, it would have only one measure for its exterior angle, and such a ratio wouldn't apply to a single polygon. Please clarify if you meant the sum or average of these ratios or something else.
Step-by-step explanation:
Let's break down each of your requests step by step.
### 1. Finding the number of sides of a regular polygon given the measure of its exterior angles in the ratio 2:3:4:9
To clarify, let's assume you are asking for the number of sides of a regular polygon where the exterior angle ratio is 2:3:4:9. This might be interpreted as the ratios of the measures of exterior angles of four different polygons.
However, if you are asking about a single regular polygon, it would have only one measure for its exterior angle, and such a ratio wouldn't apply to a single polygon. Please clarify if you meant the sum or average of these ratios or something else.
### 2. Finding each angle of the quadrilateral
Assuming the quadrilateral in question has interior angles in the ratio 2:3:4:9, we can find the measures of each angle:
1. **Sum of interior angles of a quadrilateral**:
\[
\text{Sum of interior angles} = 360^\circ
\]
2. **Total parts in the ratio**:
\[
2 + 3 + 4 + 9 = 18
\]
3. **Each part's value**:
\[
\text{Each part's value} = \frac{360^\circ}{18} = 20^\circ
\]
4. **Individual angles**:
- First angle: \(2 \times 20^\circ = 40^\circ\)
- Second angle: \(3 \times 20^\circ = 60^\circ\)
- Third angle: \(4 \times 20^\circ = 80^\circ\)
- Fourth angle: \(9 \times 20^\circ = 180^\circ\)
So, the angles of the quadrilateral are \(40^\circ, 60^\circ, 80^\circ,\) and \(180^\circ\).
### 3. Frequency distribution table for the given marks
Given marks: 21, 23, 19, 17, 12, 15, 15, 17, 17, 19, 23, 23, 21, 23, 25, 25, 21, 19, 19, 19
Construct the frequency distribution table:
| Marks | Frequency |
|-------|-----------|
| 12 | 1 |
| 15 | 2 |
| 17 | 3 |
| 19 | 5 |
| 21 | 3 |
| 23 | 4 |
| 25 | 2 |
### 4. Drawing a pie chart for the frequency distribution
To draw a pie chart, we need to find the central angle for each category:
1. **Total number of students**:
\[
\text{Total} = 20
\]
2. **Central angle for each category**:
\[
\text{Central angle} = \left( \frac{\text{Frequency}}{\text{Total}} \right) \times 360^\circ
\]
Calculate the angles:
- For 12 marks:
\[
\left( \frac{1}{20} \right) \times 360^\circ = 18^\circ
\]
- For 15 marks:
\[
\left( \frac{2}{20} \right) \times 360^\circ = 36^\circ
\]
- For 17 marks:
\[
\left( \frac{3}{20} \right) \times 360^\circ = 54^\circ
\]
- For 19 marks:
\[
\left( \frac{5}{20} \right) \times 360^\circ = 90^\circ
\]
- For 21 marks:
\[
\left( \frac{3}{20} \right) \times 360^\circ = 54^\circ
\]
- For 23 marks:
\[
\left( \frac{4}{20} \right) \times 360^\circ = 72^\circ
\]
- For 25 marks:
\[
\left( \frac{2}{20} \right) \times 360^\circ = 36^\circ
\]
### Pie Chart Representation
To draw the pie chart, use the calculated angles to represent each category. You can use a compass and protractor to draw the sectors on paper, or use software like Microsoft Excel or Google Sheets to create the pie chart digitally.
Here's a rough illustration of the pie chart:
- 12 marks: 18°
- 15 marks: 36°
- 17 marks: 54°
- 19 marks: 90°
- 21 marks: 54°
- 23 marks: 72°
- 25 marks: 36°
### Conclusion
With these calculations and methods, you can create the necessary visual and tabular representations for your data.