Lesson Notes By Weeks and Term v4 - SHS 2
SPATIAL SENSE
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SPATIAL SENSE
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Subject: Additional Mathematics
Class: SHS 2
Term: 2nd Term
Week: 3
Grade code: 2.2.1.LI.4
Strand code: 2
Sub-strand code: 1
Content standard code: 2.2.1.CS.1
Indicator code: 2.2.1.LI.4
Theme: GEOMETRIC REASONING AND MEASUREMENT
Subtheme: SPATIAL SENSE
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Imagine you are in a field and your friend tells you to stand exactly 5 meters away from a mango tree. Where could you stand? You could stand at any point in a perfect circle around the tree. This path you can walk on is a "locus." In our communities, understanding loci helps in planning things like where to build a new borehole so it's fair for two villages, or how a phone gets a signal from a cell tower. Today, we will move from just imagining these paths to describing them precisely using the language of algebra - specifically, coordinate geometry. We will learn how to turn geometric conditions into algebraic equations.
Concept 1: What is a Locus?
A locus (plural: loci) is a set of all points that satisfy one or more given conditions. It is simply the path traced by a moving point that obeys a certain rule. Analogy: Think of a goat tied to a peg with a rope of length 3 meters. The goat can move anywhere as long as the rope is tight. The path it can trace on the ground is a circle. The circle is the locus of points 3 meters from the peg.
In this lesson, we will place these paths on a Cartesian (x-y) plane and find their equations. The key tools we will use are our prior knowledge of: The distance between two points: `d = √[(x₂ - x₁)² + (y₂ - y₁)²]` The gradient (slope) of a line: `m = (y₂ - y₁)/(x₂ - x₁)` Condition 1: Locus of points at a fixed distance from a fixed point
Geometric Shape: A Circle. Condition: A point `P(x, y)` moves such that its distance from a fixed point `A(h, k)` is always a constant value, `r`. Algebraic Derivation: The condition is `|AP| = r`. Using the distance formula: `√[(x - h)² + (y - k)²] = r`. To remove the square root, we square both sides: `(x - h)² + (y - k)² = r²` This is the standard equation of a circle with centre `(h, k)` and radius `r`.