Constructions, loci and bearings
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课程笔记
Tools for Construction
- Straightedge: an idealized ruler with no markings, used only to draw a line segment between two points or extend an existing line.
- Compasses: an idealized tool for drawing circles or arcs, given a centre and a point on the circle; it can have any radius.
- In strict classical constructions, the compass is assumed to collapse when lifted, so you cannot directly transfer a distance.
- Despite the collapsing restriction, any distance can still be transferred using a multi-step procedure (the compass equivalence theorem).
- Constructions must be exact: no eyeballing, no measuring with a marked ruler, and no approximations.
- Every construction must terminate after a finite number of steps.
Constructing Triangles
- To construct a triangle you need three independent pieces of information, such as three sides (SSS), two sides and the included angle (SAS), or two angles and a side (ASA).
- Use a ruler to draw the first side to the given length.
- Use compasses to mark arcs of the given lengths from each end of the first side; their intersection gives the third vertex.
- For SAS, draw the included angle with a protractor, then mark the second side along the new ray.
- For ASA, draw the given side, then construct the two given angles at its ends; the rays meet at the third vertex.
- Always leave your construction arcs visible so the method can be checked.
Perpendicular Bisector of a Line Segment
- The perpendicular bisector of a line segment is the line that cuts it exactly in half at 90°.
- To construct it: open compasses to more than half the segment length.
- With the compass point on each end of the segment in turn, draw arcs above and below the line.
- The two points where the arcs cross are joined to give the perpendicular bisector.
- Every point on the perpendicular bisector is the same distance from both ends of the segment.
Angle Bisector
- The bisector of an angle divides it into two equal angles.
- To construct it: with the compass point on the vertex, draw an arc that crosses both arms of the angle.
- From each crossing point, draw arcs of equal radius inside the angle.
- Join the vertex to the point where those two arcs cross.
- Every point on the bisector is the same distance from both arms of the angle.
Perpendicular from a Point to a Line
- To drop a perpendicular from a point to a line, first place the compass point on the given point.
- Draw an arc that crosses the line in two places.
- From each crossing point, draw arcs of equal radius on the opposite side of the line.
- Join the given point to the point where those arcs cross; this line is perpendicular to the original line.
- The perpendicular distance from a point to a line is the shortest distance between them.
Simple Loci
- A locus (plural loci) is the set of all points that satisfy a given condition.
- The locus of points a fixed distance from a single point is a circle centred on that point.
- The locus of points a fixed distance from a line segment is a stadium shape: two parallel lines joined by semicircles at the ends.
- The locus of points a fixed distance from a straight line (infinite) is two parallel lines, one on each side.
- The locus of points equidistant from two fixed points is the perpendicular bisector of the segment joining them.
- The locus of points equidistant from two intersecting lines is the pair of angle bisectors of the angles between them.
Three-Figure Bearings
- A bearing gives a direction as an angle measured clockwise from north.
- Bearings are written as three figures, e.g. 045° or 270°.
- North is 000°, east is 090°, south is 180°, and west is 270°.
- To measure a bearing, draw a north line at the starting point and measure the clockwise angle to the direction of travel.
- Bearings are always measured from the north line, never from east or south.
Why Constructions Matter
- Straightedge-and-compass constructions allow geometric figures to be drawn with exact precision, not just approximate measurement.
- The ancient Greeks developed many constructions, including bisecting angles and constructing regular polygons with 3, 4, or 5 sides.
- Some problems, such as trisecting an arbitrary angle or doubling a cube, were later proved impossible with straightedge and compass alone.
- Pierre Wantzel proved in 1837 that these classical problems cannot be solved under the strict rules.
- Understanding constructions builds a foundation for logical reasoning and proof in geometry.
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练习题
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1.Which instrument is used to draw a circle in a ruler-and-compass construction?
Easy- AA marked ruler
- BA protractor
- CA pair of compasses
- DA set square
2.A three-figure bearing is measured clockwise from north.
EasyTrue or false?
3.Which of these is the correct three-figure bearing for a direction of 70° measured clockwise from north?
Medium- A007°
- B070°
- C700°
- D110°
4.Which construction produces the locus of points that are the same distance from two fixed points A and B?
Medium- AThe angle bisector of angle AOB
- BThe perpendicular bisector of AB
- CA circle centred at A with radius AB
- DThe line through A parallel to AB
5.Which of the following are valid ruler-and-compass constructions? (select all that apply)
Medium- ABisecting a given angle
- BConstructing the perpendicular bisector of a line segment
- CTrisecting an arbitrary angle
- DDoubling the volume of a cube
- EConstructing a perpendicular from a point to a line
6.Match each locus description to its shape.
Medium- Points a fixed distance from a single point
- Points a fixed distance from a straight line
- Points equidistant from two fixed points
- A circle
- Two parallel lines
- The perpendicular bisector of the segment joining the points
7.Put these steps for constructing the perpendicular bisector of line segment AB in the correct order.
Medium- Open compasses to more than half of AB.
- Place compass point at A and draw arcs above and below AB.
- Without changing the compass width, place compass point at B and draw arcs crossing the first arcs.
- Join the two intersection points of the arcs with a straight line.
8.Which of these angles cannot be constructed exactly using only a straightedge and compasses?
Medium- A90°
- B60°
- C45°
- D20°
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