Lesson Notes By Weeks and Term v4 - SHS 1
SPATIAL SENSE
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SPATIAL SENSE
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Subject: Additional Mathematics
Class: SHS 1
Term: 2nd Term
Week: 2
Grade code: 1.2.1.LI.6
Strand code: 2
Sub-strand code: 1
Content standard code: 1.2.1.CS.1
Indicator code: 1.2.1.LI.6
Theme: GEOMETRIC REASONING AND MEASUREMENT
Subtheme: SPATIAL SENSE
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In our daily lives, we are always trying to find the most efficient way to get from one place to another. Whether it's a farmer in the Volta Region planning the shortest irrigation pipe from a main channel to a crop bed, or a developer in Accra siting a new house as close as possible to a major road, the concept of "shortest distance" is fundamental. Today, we will explore this idea mathematically. We will discover that the shortest path from a point to a straight line is always a straight line that meets the original line at a right angle (a perpendicular line). We will learn how to calculate this distance precisely using coordinate geometry.
Part 1: Investigating the Shortest Distance (The "Why")
Let's start with a simple activity. Imagine a point P (representing your location) and a straight line L (representing a long, straight wall).
If you have to walk from P to the wall L, there are many paths you could take (PA, PB, PC, PD). Which path is the shortest? Real-Life Investigation: In the classroom, let one corner of a student's desk be the point P and the edge of the chalkboard be the line L. Use a measuring tape to measure the distance from the corner of the desk to several different points along the edge of the chalkboard. You will quickly notice that the smallest measurement you get is when the tape measure forms a perfect corner (a 90° angle) with the edge of the chalkboard. Mathematical Proof (using Pythagoras' Theorem): Look at the diagram again. The line segment PC is perpendicular to the line L, meaning it creates a right angle at point C. Now consider another path, like PD. The points P, C, and D form a right-angled triangle, with the right angle at C. The side PD is the hypotenuse. The sides PC and CD are the other two sides. According to Pythagoras' Theorem, `(PD)² = (PC)² + (CD)²`. Since `(CD)²` must be a positive value (as D is a different point from C), it means `(PD)²` must be greater than `(PC)²`. Therefore, the length PD must be longer than the length PC.
Conclusion: This is true for any point on the line other than C. The shortest possible distance from a point to a line is always the perpendicular distance. Part 2: Calculating the Shortest Distance (The "How")