construct an equilateral triangle in \( A B=B C=A C=5.9 \mathrm{~cm} \).

Constructing an equilateral triangle with each side measuring \( 5.9 \) cm can be accomplished using a compass and a straightedge. Below are the step-by-step instructions to guide you through the process: ### **Materials Needed:** - A straightedge (ruler without markings for accuracy) - A compass - A pencil - A piece of paper ### **Steps to Construct the Equilateral Triangle:** 1. **Draw the Base:** - **Step 1:** Place the straightedge on the paper and draw a straight line segment \( AB \) that is exactly \( 5.9 \) cm long. This will be the base of your equilateral triangle. 2. **Set the Compass:** - **Step 2:** Adjust your compass to a width of \( 5.9 \) cm. Ensure that the compass remains at this width throughout the construction. 3. **Draw Arcs from Point A:** - **Step 3:** Place the compass point on vertex \( A \) of the base \( AB \). - **Step 4:** Draw an arc above the base. This arc will represent all possible locations for the third vertex \( C \) that are \( 5.9 \) cm away from \( A \). 4. **Draw Arcs from Point B:** - **Step 5:** Without changing the compass width, place the compass point on vertex \( B \) of the base \( AB \). - **Step 6:** Draw another arc that intersects with the arc drawn from point \( A \). The intersection point of these two arcs will be the third vertex \( C \) of the equilateral triangle. 5. **Complete the Triangle:** - **Step 7:** Using the straightedge, draw straight lines connecting vertex \( C \) to vertices \( A \) and \( B \). Specifically, draw lines \( AC \) and \( BC \). 6. **Verify the Triangle:** - **Step 8:** Measure all three sides \( AB \), \( BC \), and \( AC \) to confirm that each is exactly \( 5.9 \) cm long. Additionally, verify that all three angles are \( 60^\circ \) each, which is characteristic of an equilateral triangle. ### **Visual Representation:** While I can't provide a visual diagram here, following the steps above with precision will result in an accurate equilateral triangle. Each step ensures that all sides and angles are equal, maintaining the properties of an equilateral triangle. ### **Tips for Accuracy:** - **Compass Stability:** Keep the compass steady and ensure that the width remains unchanged once set to \( 5.9 \) cm. - **Sharp Pencils:** Use a sharp pencil to make precise marks, especially where the arcs intersect. - **Careful Measurement:** Double-check the length of the base before proceeding to prevent cumulative errors. By following these instructions carefully, you'll successfully construct an equilateral triangle with each side measuring \( 5.9 \) cm.