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THE VISUAL LEARNING STUDIO
See the idea. Understand the connection.
35 visual explanations across 29 chapters, with topic maps throughout the library.
The chapter index will grow as additional notes are prepared. Chapters shown here are a learning collection, not a claim that the entire exam syllabus is complete.
SEE IT. CONNECT IT. EXPLAIN IT.
Understand the chapter visually
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Study the expanded notes and the credited detailed textbook lessons. The current chapter index covers core topics; it is not a claim of exhaustive coverage of every examination syllabus. Check the official syllabus and primary references.
Coordinate Geometry — lesson overview
Points and distance
A Cartesian point is represented by an ordered pair (x,y), with the coordinate order determining its position.
Distance between (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)²+(y₂−y₁)²], derived from perpendicular coordinate differences.
The midpoint is ((x₁+x₂)/2,(y₁+y₂)/2). Distance and midpoint answer different geometric questions.
Lines and slope
For two points with unequal x coordinates, slope is (y₂−y₁)/(x₂−x₁). A vertical line has undefined finite slope, not zero slope.
The point–slope form is y−y₁=m(x−x₁). Horizontal lines have zero slope and equations of the form y=c.
Nonvertical parallel lines have equal slopes; perpendicular finite slopes satisfy m₁m₂=−1, with vertical/horizontal cases treated separately.
Circles and examples
A circle with centre (h,k) and radius r has equation (x−h)²+(y−k)²=r² in Cartesian coordinates.
The distance between (0,0) and (3,4) is 5, while their midpoint is (1.5,2).
An equation’s graph must satisfy every term and condition; sketching key points can reveal algebraic sign errors.
For example, substituting a proposed point into a circle equation checks whether the point lies on that circle under the given coordinates.
