Loci and constructions

1) Loci

A locus is a path formed by a point which moves according to a rule. The plural is loci.

Running track with locus path highlighted

The runner is following a path. This path is a locus.

Locus around a point

The hands of a clock move around the clock and create a locus. The tip of each hand is always the same distance – equidistant – from the centre of the clock.

The locus the hands create is a circle.

Clock face with locus path highlighted

A locus of points at equal distance around a point is a circle.

A pair of compasses must be used to create a locus around a point.

Question 1

Farmer Smith has tied a cow around a post on a rope 4 m long. What the locus of the cow as it moves around the post?

Path of cow highlighted

Answer: The locus is a circle 4 m from the post.

Cow following a locus path with a radius of 4m

Locus from a line

A locus does not have to be a line – it can also be an area.

A flowerbed runs along the grass between A and B. The edge of the flowerbed is 1 m from the grass. How would you draw an accurate diagram showing the flowerbed, using a scale of 1 cm:1 m?

Empty flower bed

Draw a line parallel to AB 1 cm from AB. Shade in the area between this line and the line AB.

Flower bed with area filled in with flowers

The edge of the flowerbed is the locus from the line AB, because it is at a set distance from the line.

Question

Visitors must stand at least 2 m away from the walls of the monkey enclosure at the zoo. How would you draw an accurate diagram to show where the visitors must not stand, using a scale of 1 cm:1 m?

Zoo enclosure (DEFG)

Answer: Draw four lines parallel to each side of the rectangle, 2 cm away. Use a compass to draw rounded corners at D, E, F and G.

Zoo enclosure (DEFG) with 2m surrounding locus highlighted

The straight line of the locus is parallel to the edges of the rectangle, because they are at a set distance from the edges. At each corner the locus is a quarter circle drawn from the fixed points D, E, F and G.

Visitors must not stand in the area between the rectangles. Shade this in.

Area created by locus, shaded blue

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