lesson
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Imagine two rival pizza shops in town want to build a shared delivery hub that is exactly the same distance from both stores.
There is not just one fair spot—there is an entire straight boundary line of infinite locations that work.
The Perpendicular Bisector
A midpoint divides a segment into two equal halves, and a line is perpendicular if it crosses at a 90∘ right angle.
Combining these ideas creates the perpendicular bisector—a line that cuts a segment in half at a crisp 90∘ angle.
📊Interactive/animated geometric diagram showing a horizontal line segment AB with midpoint M. A vertical line L passes through M with a right-angle square symbol at M and tick marks showing AM = MB. A point P sits on line L with dashed lines connecting P to A and P to B. Distances PA and PB are labeled with equal measurement badges (e.g., '10 cm' and '10 cm'). Point P smoothly glides up and down along line L, dynamically updating the distance labels in sync so both always stay identical. Clean modern card layout, white background #ffffff, dark navy text #1e2945, accent blue lines #2563eb, coral endpoints #f43f5e.
Why does sliding point P anywhere up or down that vertical line keep the distances to A and B perfectly equal?
The Perpendicular Bisector Theorem
Over 2,300 years ago, Greek mathematician Euclid recorded this property in his Elements to construct points that are equidistant—meaning precisely equal in distance—from two fixed endpoints.
The Perpendicular Bisector Theorem states that if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment.
The Proof
To prove this, connect any point P on the bisector to endpoints A and B, which splits the figure into two right triangles: △PMA and △PMB.
📊Step-by-step visual proof card showing segment AB and perpendicular bisector PM creating two adjacent right triangles: triangle PMA (tinted light blue) and triangle PMB (tinted light purple). Visual callouts highlight the 3 matching parts: 1) AM congruent to BM (marked with single tick marks), 2) angle PMA congruent to angle PMB = 90 degrees (marked with right angle symbols), and 3) shared side PM congruent to PM (marked with double tick marks). A central banner reads: 'Triangle PMA is congruent to Triangle PMB by SAS Congruence' and arrows point to PA = PB by CPCTC. Clean typography, #1e2945 text, subtle shadows.