Constructing figures from measurements

Use a ruler for lengths and straight lines, a protractor for measured angles, compasses for equal distances, and a set square for perpendicular or parallel lines when permitted. A compass-and-straightedge construction establishes a relationship using arcs; measuring an approximate midpoint is not that construction.

Example

To construct \(\triangle ABC\) with \(AB=6\text{ cm}\), \(AC=5\text{ cm}\), \(BC=4\text{ cm}\): draw \(AB\); draw an arc of radius 5 cm centred at \(A\), and one of radius 4 cm centred at \(B\); label their intersection \(C\) and join \(AC\) and \(BC\). Retain the arcs.

The arcs encode \(AC=5\) and \(BC=4\). A reflected intersection below \(AB\) gives a congruent triangle; choose the requested side. For two sides and an included angle, construct the angle first and mark the two given lengths on its arms. For two angles and an included side, draw the side, construct the two endpoint angles, and extend their rays to meet.

Check feasibility before drawing: the longest side of a triangle must be shorter than the sum of the other two. Use the stated data to build quadrilaterals as connected triangles or with the required parallel/perpendicular sides. Never measure a sketch labelled “not drawn to scale”.