Pivot a Line: Meeting a Segment, Halving an Area
A line through a fixed point turns like a hand on a dial. When the whole segment lies on one side of the vertical line through the fixed point, the slope to a point of the segment runs from one endpoint's slope to the other's, so 'meets the segment' is the slope range between the endpoints, ends included. Halving an area works the same way: decide which side the line must cross, then fix the crossing point by area. A line through the centre of a centrally symmetric figure always halves it.
grade 7–8 FunctionsGeometryExpressions Draw a DiagramVisualize Spatial Relationships
Progression (7)
g7 1. A line through the origin sweeps between endpoints RatiosNumber systemExpressions
g7 2. Decide which side the halving line crosses RatiosGeometryExpressions
g8 3. Rotate a line about its fixed point FunctionsExpressions
g8 4. Find which side the bisector hits first GeometryFunctions
g8 5. Area bisectors of a square pass through its center FunctionsGeometry
g8 6. Pivot a line past a triangle's extreme vertices FunctionsExpressions
g8 7. Bisect a pentagon by a line from one vertex FunctionsGeometry