Constructions, loci and bearings

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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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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.
Easy
True 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°

Plus de questions

9.Which statements about loci are true? (select all that apply)
Medium
- AA locus is a set of points that satisfy a given condition.
- BThe locus of points a fixed distance from a point is a circle.
- CThe locus of points a fixed distance from a line is a single straight line.
- DThe locus of points equidistant from two intersecting lines is a pair of angle bisectors.
10.The angle bisector of an angle is the locus of points equidistant from the two arms of the angle.
Easy
True or false?
11.Match each construction to the tool or method used.
Medium
- Drawing a straight line through two points
- Drawing a circle with a given centre and radius
- Constructing a 60° angle
- Straightedge
- Compasses
- Constructing an equilateral triangle
12.Put these steps for constructing the bisector of angle ABC in the correct order.
Medium
- With centre B, draw an arc crossing both arms of the angle.
- Label the intersection points on the arms as P and Q.
- With centres P and Q, draw arcs that cross inside the angle.
- Draw a straight line from B through the crossing point of the two arcs.
13.Match each construction task to the tool that is essential for it.
Medium
- Drawing a circle with a given centre and radius
- Drawing a straight line segment between two marked points
- Marking a point at a measured distance from another point
- Drawing an arc of a given radius from a point
- Compasses
- Straightedge only
- Ruler with markings
- Compasses set to that radius
14.Put the steps for constructing a triangle with side lengths AB = 7 cm, BC = 5 cm and AC = 4 cm in the correct order.
Medium
- Draw the base line AB of length 7 cm.
- Set the compasses to radius 5 cm and draw an arc centred on B.
- Set the compasses to radius 4 cm and draw an arc centred on A.
- Mark the point C where the two arcs cross.
- Draw the line segments AC and BC to complete the triangle.
15.A point P is to be constructed so that it is 3 cm from a fixed point A and also 3 cm from a fixed point B, where AB = 5 cm. Which statements are true? (select all that apply)
Medium
- AP lies on the circle of radius 3 cm centred on A.
- BP lies on the perpendicular bisector of AB.
- CP lies on the circle of radius 3 cm centred on B.
- DP lies on the line AB.
- EThere are two possible positions for P.
16.A three-figure bearing is always measured clockwise from north.
Easy
True or false?
17.Which construction gives the locus of points that are the same distance from two intersecting lines?
Medium
- AThe angle bisector of the angle between the lines
- BThe perpendicular bisector of a segment joining the lines
- CA circle centred on the intersection point
- DA line parallel to one of the lines
18.Which of the following can be constructed exactly using only a straightedge and compasses? (select all that apply)
Hard
- AThe perpendicular bisector of a line segment
- BThe bisector of any given angle
- CA regular 17-sided polygon
- DA regular 7-sided polygon
- EA square with the same area as a given circle
19.Put the steps for constructing the perpendicular from a point P to a line l in the correct order.
Medium
- With centre P, draw an arc that cuts line l at two points, A and B.
- With centre A, draw an arc on the opposite side of l from P.
- With centre B and the same radius, draw an arc crossing the previous arc at Q.
- Draw the line through P and Q; this is the perpendicular to l.
20.A ship sails on a bearing of 200°. It then turns to a bearing of 110°. Through how many degrees has it turned, and in which direction?
Medium
- A90° anticlockwise
- B90° clockwise
- C270° anticlockwise
- D310° clockwise
21.A straightedge is used in two ways only. Which pair of actions is allowed?
Easy
- ADraw a line segment between two points, or extend an existing line segment
- BMeasure a length, or draw a circle
- CDraw a circle, or mark a point at a given distance
- DCopy an angle, or bisect a segment
22.In a ruler-and-compass construction, which instrument is used to draw a circular arc?
Easy
- ACompasses
- BStraightedge
- CProtractor
- DSet square
23.In classical construction, a straightedge is allowed to have markings on it.
Easy
True or false?
24.A construction is only acceptable if it terminates after a finite number of steps.
Easy
True or false?
25.Which of these operations is NOT permitted in a straightedge-and-compass construction?
Medium
- AMeasuring a length with a marked ruler
- BDrawing a line through two constructed points
- CDrawing a circle with a given centre and radius
- DExtending a line segment
26.Which of the following are true about the idealized compass used in classical construction? (select all that apply)
Medium
- AIt has no markings
- BIt can have an arbitrarily large radius
- CIt may be assumed to collapse when lifted from the page
- DIt can be used to measure angles directly
- EIt can only draw circles of radius less than 10 cm
27.A construction must be mathematically exact. Which statement best explains what this means?
Medium
- ANo 'eyeballing' distances and no using markings on a ruler are allowed
- BThe drawing must look accurate when checked by eye
- CThe construction may use a marked ruler as long as it is accurate
- DThe construction may be an approximation that gets closer and closer
28.Which of these lengths can be constructed from a unit length using only straightedge and compasses?
Medium
- A\√{2}
- B\√[3]{2}
- C\π
- D\√[5]{2}
29.A length is constructible if and only if it represents a constructible number. Which condition describes a constructible number?
Medium
- AIt can be written using the four basic arithmetic operations and the extraction of square roots, but no higher-order roots
- BIt is any positive integer
- CIt can be written using cube roots and square roots
- DIt is any number that can be measured with a ruler
30.Which of the following ancient construction problems were proved impossible with straightedge and compasses alone? (select all that apply)
Medium
- ATrisecting an arbitrary angle
- BDoubling the volume of a cube
- CBisecting an arbitrary angle
- DConstructing a square with the same area as a given circle
- EConstructing a regular pentagon
31.Pierre Wantzel proved in 1837 that certain constructions are impossible. Which pair did he prove impossible?
Medium
- ATrisecting an arbitrary angle and doubling the volume of a cube
- BBisecting an angle and drawing a perpendicular bisector
- CConstructing a regular pentagon and a regular hexagon
- DAdding and subtracting lengths
32.The ancient Greeks could construct regular polygons with certain numbers of sides. Which of these did they construct?
Medium
- A3, 4 and 5 sides
- B7, 9 and 11 sides
- C6, 10 and 14 sides
- DAll whole numbers of sides
33.Gauss contributed to the study of which regular polygon, showing in 1796 that it could be constructed?
Medium
- AA regular 17-sided polygon
- BA regular 7-sided polygon
- CA regular 9-sided polygon
- DA regular 11-sided polygon
34.A ship sails 8 km on a bearing of 090°, then 6 km on a bearing of 180°. The ship is now at point C. What is the direct distance from the starting point to C?
Medium
- A10 km
- B14 km
- C2 km
- D48 km
35.A ship sails 8 km on a bearing of 090°, then 6 km on a bearing of 180°. What is the three-figure bearing of C from the starting point? Give your answer to the nearest degree.
Medium
- A037°
- B053°
- C127°
- D217°
36.A ship sails 8 km on a bearing of 090°, then 6 km on a bearing of 180°. What is the three-figure bearing of the starting point from C?
Medium
- A307°
- B127°
- C217°
- D053°
37.A point P is 5 cm from a fixed point A. Which single construction gives the locus of points that are both 5 cm from A and 3 cm from a fixed line l?
Medium
- Athe points where the circle centre A radius 5 cm meets the pair of parallel lines 3 cm from l
- Bthe points where the circle centre A radius 5 cm meets the perpendicular bisector of A and l
- Cthe points where the perpendicular bisector of A and the pair of parallel lines 3 cm from l meet
- Dthe points where the circle centre A radius 3 cm meets the pair of parallel lines 5 cm from l
38.A point P is 5 cm from a fixed point A and 3 cm from a fixed line l. The circle centre A radius 5 cm and the pair of parallel lines 3 cm from l intersect in four points. Which statements about these points are true? (select all that apply)
Hard
- AEach point is exactly 5 cm from A.
- BEach point lies on the perpendicular bisector of A and l.
- CEach point is exactly 3 cm from l.
- DThere are exactly two such points.
- EEach point is exactly 5 cm from l.
39.A point P moves so that it is always the same distance from two fixed points A and B. Which locus does P trace?
Medium
- Athe perpendicular bisector of AB
- Bthe circle with centre A and radius AB
- Cthe pair of lines parallel to AB at a fixed distance
- Dthe angle bisector of angle APB
40.A point P moves so that it is always the same distance from two fixed intersecting lines l and m. Which locus does P trace?
Medium
- Athe two angle bisectors of the angles formed by l and m
- Bthe perpendicular bisector of the segment joining l and m
- Cthe pair of lines parallel to l at a fixed distance
- Dthe circle with centre at the intersection of l and m
41.A bearing is given as 45°. Write it as a three-figure bearing. Give your answer as a three-digit number.
Medium
42.Complete the definition: A three-figure bearing is measured ___ from ___.
Medium
A three-figure bearing is measured ____ from ____.
43.A ship sails on a bearing of 120°. Through how many degrees must it turn clockwise to sail on a bearing of 300°?
Medium
44.Complete the sentence about bearings.
Medium
A bearing is always written using ____ digits, for example 045°.
45.A point Q is 6 cm from a fixed point P. What is the radius, in centimetres, of the locus of all points 6 cm from P?
Medium
46.What is the name of the locus of points that are a fixed distance from a given straight line?
Medium
47.A point P is 4 cm from a fixed point O. A locus is drawn of all points that are 4 cm from O. State the radius of this locus, in centimetres.
Easy
48.The locus of all points that are 5 cm from a given straight line is a pair of parallel lines. State the perpendicular distance, in centimetres, between each of these lines and the given line.
Medium
49.A ship is on a bearing of 075°. It turns anticlockwise through 130°. What is its new three-figure bearing? Give your answer as a three-digit number of degrees.
Medium
50.A walker faces a bearing of 250° and turns clockwise until facing a bearing of 030°. Through how many degrees has the walker turned?
Medium
51.Complete the statement about bearings.
Easy
Three-figure bearings are always measured in a ____ direction from ____.
52.Complete the sentence about constructing a perpendicular bisector.
Medium
To construct the perpendicular bisector of AB, open the compasses to a radius ____ than half of AB and draw arcs from both A and B.
53.Explain why a straightedge used in classical construction has no markings.
Hard
54.Explain why it is impossible to trisect an arbitrary angle using only a straightedge and compasses.
Hard
55.Explain why it is impossible to construct a regular 7-sided polygon using only a straightedge and compasses.
Hard
56.Explain why a straightedge used in a classical construction must have no markings on it.
Hard
57.A length is constructible if and only if it represents a constructible number. Describe the condition that makes a number constructible.
Hard
58.A bearing is given as 45°. Write it as a three-figure bearing. Give your answer as a three-digit number.
Medium
- A40
- B50
- C55
- D45
59.A ship sails on a bearing of 120°. Through how many degrees must it turn clockwise to sail on a bearing of 300°?
Medium
- A180°
- B216°
- C198°
- D162°
60.A point Q is 6 cm from a fixed point P. What is the radius, in centimetres, of the locus of all points 6 cm from P?
Medium
- A8 cm
- B5 cm
- C6 cm
- D7 cm
61.A point P is 4 cm from a fixed point O. A locus is drawn of all points that are 4 cm from O. State the radius of this locus, in centimetres.
Easy
- A6 cm
- B3 cm
- C4 cm
- D5 cm
62.The locus of all points that are 5 cm from a given straight line is a pair of parallel lines. State the perpendicular distance, in centimetres, between each of these lines and the given line.
Medium
- A7 cm
- B5 cm
- C6 cm
- D4 cm
63.A ship is on a bearing of 075°. It turns anticlockwise through 130°. What is its new three-figure bearing? Give your answer as a three-digit number of degrees.
Medium
- A274 °
- B367 °
- C336 °
- D305 °
64.A walker faces a bearing of 250° and turns clockwise until facing a bearing of 030°. Through how many degrees has the walker turned?
Medium
- A126 °
- B168 °
- C154 °
- D140 °

Notes de révision

## Constructions, Loci and Bearings ### Constructions with ruler, protractor and compasses - Use a **ruler** for straight lines, a **protractor** for angles, and **compasses** for arcs and circles. - **Constructing triangles**: given sides and angles, draw a base line, then use compasses (for sides) or protractor (for angles) to locate the third vertex. Check with the given measurements. - **Perpendicular bisector of a line**: set compasses to more than half the line length, draw arcs from each end, then join the two intersection points. Every point on it is equidistant from the two ends. - **Angle bisector**: from the vertex, draw an arc crossing both arms; from those crossings, draw equal arcs to meet; join the vertex to that meeting point. It cuts the angle exactly in half. - **Perpendicular from a point to a line**: from the point, draw an arc crossing the line twice; from those crossings, draw equal arcs to meet; join the point to that meeting point. ### Loci - A **locus** is the set of all points satisfying a rule. - Points a fixed distance from a point: a **circle**. - Points a fixed distance from a line: two parallel lines, one on each side. - Points equidistant from two points: the **perpendicular bisector** of the segment joining them. - Points equidistant from two lines: the **angle bisector**. ### Three-figure bearings - Bearings are measured **clockwise from north**. - Always give **three figures**: e.g. 045°, 120°, 310°. - North is 000°, east is 090°, south is 180°, west is 270°. - To find a back bearing, add or subtract 180°. ### Key facts - Straightedge-and-compass constructions use only an unmarked straightedge and a compass; measurements are not allowed. - Each construction must be exact and finish in a finite number of steps. - A length is constructible if it can be made using +, −, ×, ÷ and square roots only. - Angles can be bisected, but trisecting an arbitrary angle is impossible with straightedge and compass alone. ### Exam tips - Leave all construction arcs visible. - Label points and angles clearly. - For bearings, start at north and measure clockwise; write three digits. - Check that your construction matches the given lengths and angles.

Straightedge-and-compass construction: Construction of lengths, angles and figures using only an unmarked straightedge and a compass.
Perpendicular bisector: The line that cuts a segment in half at right angles; every point on it is equidistant from the segment's endpoints.
Angle bisector: The line that divides an angle into two equal angles.
Locus: The set of all points that satisfy a given rule.
Three-figure bearing: An angle measured clockwise from north, written with three digits, e.g. 045°.
Constructible number: A number that can be written using +, −, ×, ÷ and square roots only.

Cartes mémo

What is the difference between a ruler and a straightedge in geometric construction?

A straightedge is an idealized ruler with only one edge, no markings and infinite length. It can only draw a line segment between two points or extend an existing line.

What is the only thing a straightedge may be used for in a construction?

Drawing a line segment between two points, or extending an existing line segment. No measuring or marking is allowed.

What is an idealized compass assumed to be able to do?

Draw circles or circular arcs with any radius, given a centre point and a point on the circle. It has no maximum or minimum radius and no markings.

What does it mean that an idealized compass may 'collapse'?

When lifted from the page it loses its setting, so it cannot directly transfer a distance. By the compass equivalence theorem, no construction power is lost by this.

Why must a straightedge-and-compass construction be exact rather than approximate?

So that the construction can be proven to be exactly correct. 'Eyeballing' distances or using ruler markings is not permitted, and the construction must terminate in a finite number of steps.

What is the perpendicular bisector of a line segment?

The line that cuts the segment in half at a right angle (90°). Every point on it is the same distance from both ends of the segment.

How do you construct the perpendicular bisector of a line segment AB?

Open compasses to more than half of AB. Draw arcs centred on A and on B on both sides of the line. Join the two points where the arcs cross.

What is the angle bisector of an angle?

The line that divides the angle into two equal parts. Every point on it is the same distance from both arms of the angle.

How do you construct the bisector of an angle?

With centre at the vertex, draw an arc cutting both arms. From those two crossing points, draw arcs that cross inside the angle. Join the vertex to that crossing point.

How do you construct the perpendicular from a point to a line?

With centre at the point, draw an arc cutting the line twice. From those two points draw equal arcs to cross on the other side. Join the point to that crossing.

What is a locus?

The set of all points that satisfy a given rule or condition. For example, all points a fixed distance from a given point.

What is the locus of points a fixed distance r from a single point P?

A circle of radius r with centre P.

What is the locus of points a fixed distance d from a straight line?

Two straight lines parallel to the given line, one on each side, each at distance d from it.

What is the locus of points equidistant from two fixed points A and B?

The perpendicular bisector of the line segment AB.

What is the locus of points equidistant from two intersecting lines?

The two angle bisectors of the angles formed by the lines.

What is the locus of points equidistant from two parallel lines?

A single straight line halfway between them, parallel to both.

What are the three things needed to construct a triangle from given data?

A ruler, a protractor and compasses. You need enough information, such as three sides (SSS) or two sides and the included angle (SAS).

How do you construct a triangle given all three side lengths?

Draw one side with a ruler. With compasses set to the second length, draw an arc from one end; set to the third length and draw an arc from the other end. Join the crossing point to both ends.

How do you construct a triangle given two sides and the included angle?

Draw one side, measure the given angle at one end with a protractor, draw the second side along that ray, then join its end to the other end of the first side.

In what direction are three-figure bearings measured?

Clockwise from north. They are always written with three figures, from 000° to 360°.

What is the bearing of due east, and of due south?

Due east is 090° and due south is 180°. Due west is 270°.

How do you find the back bearing of a point from a given bearing?

Add 180° if the bearing is less than 180°, or subtract 180° if it is 180° or more. The result is the bearing back to where you started.

How is a bearing of a point measured and stated?

Measure the angle clockwise from the north line at the starting point to the line joining the two points, then write it as three figures, e.g. 047°.

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