How do you construct a triangle given three sides?

To construct a triangle given three sides, use a ruler and compass to draw each side accurately.

First, label the three sides as \(a\), \(b\), and \(c\). Start by drawing the longest side, say \(a\), using a ruler. This will be the base of your triangle. Mark the endpoints of this line as points \(A\) and \(B\).

Next, set your compass to the length of side \(b\). Place the compass point on \(A\) and draw an arc above the line segment \(AB\). This arc represents all possible points that are \(b\) units away from \(A\).

Then, adjust your compass to the length of side \(c\). Place the compass point on \(B\) and draw another arc that intersects the first arc. The intersection of these two arcs is point \(C\), which is the third vertex of the triangle.

Finally, use a ruler to draw line segments \(AC\) and \(BC\) to complete the triangle. Ensure that the lengths of these segments match the given sides \(b\) and \(c\).

By following these steps, you can accurately construct a triangle with the given side lengths. This method ensures that the triangle adheres to the properties of Euclidean geometry, where the sum of the lengths of any two sides must be greater than the length of the remaining side.