Inaugurated by IN-SPACe
ISRO Registered Space Tutor
S1-SA3-0895
What is the Locus of Points Equidistant From a Line?
Grade Level:
Class 5
Geometry, Computing, Physics, AI
Definition
What is it?
The locus of points equidistant from a line is another line that runs parallel to the first line. 'Locus' means a set of all points that satisfy a given condition. So, if you have a line, and you find all the points that are the same distance away from it, those points will form a new straight line.
Simple Example
Quick Example
Imagine you are standing on the cricket pitch boundary line. If you walk straight, always keeping exactly 5 meters away from the boundary line, the path you trace will be a straight line parallel to the boundary line. This path is the locus of points 5 meters equidistant from the boundary line.
Worked Example
Step-by-Step
PROBLEM: Draw the locus of points that are 3 cm away from a straight line segment AB, which is 8 cm long.
---STEP 1: Draw a straight line segment AB, 8 cm long, on your paper.
---STEP 2: At point A, draw a perpendicular line segment upwards, 3 cm long. Mark its endpoint as C.
---STEP 3: At point B, draw a perpendicular line segment upwards, 3 cm long. Mark its endpoint as D.
---STEP 4: Draw a straight line connecting point C to point D. This line CD is parallel to AB.
---STEP 5: Repeat steps 2-4, but draw the perpendicular lines downwards from A and B. Mark these new endpoints E and F, and connect them to form line EF.
---ANSWER: The locus of points 3 cm from line segment AB consists of two parallel line segments, CD and EF, each 3 cm away from AB, along with two semi-circles at the ends connecting CD and EF if we consider points equidistant from the line segment's ends.
Why It Matters
Understanding loci helps in designing roads and buildings efficiently, ensuring things are spaced correctly. Engineers use this to plan safe distances for railway tracks or to position antennas for best signal coverage. It's also fundamental in computer graphics and robotics for path planning.
Common Mistakes
MISTAKE: Thinking the locus is just one line. | CORRECTION: For a given line, there are usually two lines equidistant from it – one on each side, unless a specific side is mentioned.
MISTAKE: Drawing lines that are not parallel. | CORRECTION: The key condition is 'equidistant', meaning the distance is always the same. This only happens if the new line is parallel to the original line.
MISTAKE: Confusing distance from a line with distance from a point. | CORRECTION: Distance from a line means the shortest distance, which is always measured along a perpendicular line. Distance from a point creates a circle.
Practice Questions
Try It Yourself
QUESTION: A road is 10 meters wide. A cyclist always rides exactly 2 meters from the left edge of the road. What is the locus of the cyclist's path? | ANSWER: A straight line parallel to the left edge of the road, 2 meters away from it.
QUESTION: On a map, a river is shown as a straight line. If a new walking path needs to be built so that it is always 50 meters from the river, describe the path. | ANSWER: The path will be two straight lines, parallel to the river, one on each side, both 50 meters away from the river.
QUESTION: Draw a line segment PQ that is 6 cm long. Now, draw the locus of all points that are exactly 2 cm from the line segment PQ. (Hint: Consider points above, below, and at the ends). | ANSWER: Two parallel line segments, each 2 cm from PQ (one above, one below), and two semi-circles at the ends of PQ with a radius of 2 cm, connecting the parallel lines.
MCQ
Quick Quiz
What shape is formed by all points that are exactly 7 cm away from a very long straight railway track?
A circle
A point
Two straight lines parallel to the track
A single straight line perpendicular to the track
The Correct Answer Is:
C
The condition 'equidistant from a line' means the points form parallel lines. Since it doesn't specify which side, there will be two such lines, one on each side of the railway track.
Real World Connection
In the Real World
When city planners design new infrastructure, like laying water pipelines or electricity cables, they often need to ensure these are a specific safe distance from existing structures or roads. They use the concept of locus to mark out these parallel paths, ensuring safety and compliance with regulations.
Key Vocabulary
Key Terms
LOCUS: A set of all points that satisfy a given condition. | EQUIDISTANT: Being the same distance from something. | PARALLEL LINES: Lines that are always the same distance apart and never meet. | PERPENDICULAR: At an angle of 90 degrees to a given line or surface.
What's Next
What to Learn Next
Great job understanding the locus from a line! Next, you can explore 'What is the Locus of Points Equidistant From a Point?'. This will help you see how changing the 'from what' (line vs. point) changes the shape of the locus, which is super interesting!
