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Loci and constructions - AQAConstructions: bisecting lines and angles

Loci are a set of points with the same property. Loci can be used to accurately construct lines and shapes. Bearings are three figure angles measured clockwise from North.

Part of MathsGeometry and measure

Constructions: bisecting lines and angles

Constructing a perpendicular bisector

A plane flies at equal distance between two control towers.

Plane following a locus line, between two control towers

The of the plane is the bisector of the two towers. When a line is divided into two equal lengths, it has been .

Example

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Draw the perpendicular bisector of the points X and Y
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Draw a line between the points
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Place the compass on point X. Open the pair of compasses until it is over halfway along the line. Draw an arc that crosses the line
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Place the compass on point Y without changing the width of your compass. Draw another arc that crosses the line
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Use a ruler to join the points A and B
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The line AB crosses the line XY at a right angle and divides the line XY into two equal lengths

Constructing the bisector of an angle

Example 1

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A path is constructed so it is at equal distance from the two edges of the field, JM and LM
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The locus of the path is the angle bisector of the angle LMJ. The path has separated the angle into two equal angles

Example 2

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Bisect the angle V
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Place the compass point at V. Draw an arc to cross the two lines
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Place the compass point at A. Draw an arc in between the two lines
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Place the compass point at B without changing the width of your compass. Draw an arc that crosses the arc from point A
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Join the point C to V using a ruler
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The angles AVC and BVC are equal

Solving problems using constructions

Problems can involve intersecting loci. It may be necessary to use several constructions to locate a region.

Example

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Two lifeboats are searching for a boat in distress. The boat is more than 40 m from the coastal path. It is closer to boat W than boat Z. Shade the region in which the boat lies. Use a scale of 1 cm:10 m
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Draw a line parallel to the path 4 cm from the path. Shade in the area which is more than 4 cm from the path
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Draw a line between Z and W. Construct the perpendicular bisector of this line. Shade in the region which is closer to W than Z. The boat lies where the two regions overlap

Constructing triangles

Constructions between points and lines

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