Draw a triangle of circumradius 4 centimetres and two of the angles are 60 and 80

To draw a triangle with a circumradius of 4 cm and two angles of 60° and 80°, we need to find the lengths of its sides using the Law of Sines. Here’s the process to draw the triangle step-by-step:

### Step-by-step Construction:

1. **Understand the Given Information:**
   - Circumradius (R) = 4 cm
   - Angles: \( \angle A = 60^\circ \), \( \angle B = 80^\circ \)
   - Since the sum of the angles in a triangle is 180°, the third angle \( \angle C = 180^\circ - 60^\circ - 80^\circ = 40^\circ \).

2. **Use the Law of Sines:**
   The Law of Sines relates the sides of a triangle to the sine of its angles:
   \[
   \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R
   \]
   Where:
   - \( a \), \( b \), and \( c \) are the lengths of the sides opposite to angles \( A \), \( B \), and \( C \) respectively.
   - \( R = 4 \) cm.

   Using the above relation:
   \[
   a = 2R \cdot \sin A = 2 \times 4 \times \sin(60^\circ)
   \]
   \[
   b = 2R \cdot \sin B = 2 \times 4 \times \sin(80^\circ)
   \]
   \[
   c = 2R \cdot \sin C = 2 \times 4 \times \sin(40^\circ)
   \]

3. **Calculate Side Lengths:**
   \[
   a = 8 \times \sin(60^\circ) = 8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3} \, \text{cm} \approx 6.93 \, \text{cm}
   \]
   \[
   b = 8 \times \sin(80^\circ) \approx 8 \times 0.985 = 7.88 \, \text{cm}
   \]
   \[
   c = 8 \times \sin(40^\circ) \approx 8 \times 0.643 = 5.14 \, \text{cm}
   \]

4. **Draw the Triangle:**
   - Draw a line segment \( BC \) of length \( c = 5.14 \) cm.
   - At point \( B \), use a protractor to construct an angle of 80°.
   - At point \( C \), construct an angle of 60°.
   - The two rays from points \( B \) and \( C \) will intersect at point \( A \).
   - Connect points \( A \) to \( B \) and \( A \) to \( C \) to complete the triangle.

### Summary
You should now have a triangle with side lengths approximately 6.93 cm, 7.88 cm, and 5.14 cm, with the given angles and a circumradius of 4 cm.

Would you like more details on any of the steps?

### Further Related Questions:
1. How do you calculate the circumradius of a triangle given its side lengths?
2. What is the Law of Sines and how is it applied in triangle problems?
3. How to construct a triangle given its circumradius and all three angles?
4. How does the circumcircle relate to the triangle's properties?
5. What are the other ways to solve triangles with given angles and circumradius?

**Tip:** When solving triangle problems, always remember the relationship between angles and side lengths using trigonometric laws.

Learn how to construct a triangle with a circumradius of 4 cm and angles of 60° and 80° using the Law of Sines. This problem is ideal for students in Grades 9-10, covering key trigonometric concepts and step-by-step construction.

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