Lesson 4.4
The Bisector Theorems
Focus: search-and-rescue beacon — water tower — angle bisector and perpendicular bisector theorems — constructions as proofs
This lesson turns constructions into measurement tools. Students have already built angle bisectors and perpendicular bisectors — they know how to make them. Today they discover what membership in those constructions actually guarantees: a precise relationship between a point on a bisector and the distances to the lines or points it bisects.
Both theorems emerge from a single real-world framing — where must something be placed to be equally close to two things? — and both are proven using triangle congruence tools from Unit 3. The angle bisector theorem comes from the beacon problem (equal perpendicular distance to two lines). The perpendicular bisector theorem comes from the water tower problem (equal direct distance to two points). The scenarios are genuinely different, and that distinction is worth naming when both results emerge.
Two new definitions are formalized in consolidation: perpendicular distance (the shortest distance from a point to a line, measured along the perpendicular) and equidistant (equal perpendicular distance from two lines, or equal direct distance from two points). Both will have been used informally during the tasks.
The perpendicular bisector construction is a derived result here rather than a standalone procedure. Students can construct it using only the altitude technique from Lesson 4.2: the altitude from the apex of an isosceles triangle lands on the midpoint of the base (proven by CPCTC in the altitude proof) and is perpendicular to it. To construct the perpendicular bisector of any segment AB, treat AB as the base of an isosceles triangle by finding two points equidistant from A and B, then connect them. Students already know every step of this.
Close with the structural observation: both bisector theorems make the same fundamental claim. A bisector is not just a line that divides something in half — it is the precise set of all points satisfying an equal-distance condition. The angle bisector is the set of all points equidistant from two lines. The perpendicular bisector is the set of all points equidistant from two points. The constructions build these lines; the theorems tell you what membership in them means. This is the first time in the course that students encounter a geometric object defined as a set of points satisfying a condition rather than as a construction procedure.