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Dành cho nhà giáo dục: slide bài học, ghi chú ôn tập sẵn dùng cho Constructions, loci and bearings (Mathematics, Geometry and measures) — dùng trong bài giảng của bạn, hoặc chạy chủ đề như một hoạt động lớp học tương tác để người học chơi như một trò chơi trực tiếp.
Use the tools specified. Ruler-and-compass constructions exclude a protractor; bearings and some accurate triangle drawings allow one.
Leave all construction arcs visible – they are required for full marks.
Perpendicular-bisector endpoint arcs must match. For an angle bisector, the two arcs from the arm points must match each other; their radius need not match the initial vertex arc.
For bearings, always measure clockwise from north and write as three figures (e.g., 065°, not 65°).
Perpendicular Bisector of a Line Segment
To construct the perpendicular bisector of segment AB, set compass radius greater than half AB.
Draw arcs from A and B on both sides of the line; the arcs intersect at two points.
Join the two intersection points with a straight line – this is the perpendicular bisector.
Every point on this line is equidistant from A and B.
Perpendicular bisector of AB
Perpendicular from a Point to a Line
Given a line and a point P (not on the line), place compass point at P and draw an arc cutting the line at two points.
From each of those two intersection points, draw arcs below the line (or on the opposite side) with the same radius.
Join P to the intersection of these two arcs – this is the perpendicular from P to the line.
The foot of the perpendicular is the point where it meets the line.
Perpendicular from P to a line
Angle Bisector
To bisect angle ABC, place compass at vertex B and draw an arc crossing both arms (BA and BC).
From the two arm-crossing points, draw equal-radius arcs large enough to intersect inside the angle. Their radius can differ from the initial vertex arc.
Draw a straight line from B through the intersection of the arcs – this is the angle bisector.
Every point on the bisector is equidistant from the two arms of the angle.
Bisect angle ABC
Constructing Triangles
SSS (three sides): draw one side, then use compasses to mark arcs from each end with the other two lengths; the intersection gives the third vertex.
SAS (two sides and included angle): draw the base, measure the angle with a protractor (if allowed) or use a compass construction, then mark the second side length.
ASA (two angles and included side): draw the base, construct the two angles at each end using a protractor or compass, and extend the rays to meet.
Always leave construction lines visible to show your method.
Construct a triangle from three sides
Loci: Fixed Distance from a Point
The locus of points at a fixed distance from a point is a circle centred on that point.
The radius equals the fixed distance.
To construct it, set compass to the required radius and draw a full circle.
Fixed distance from a point
Loci: Fixed Distance from a Line
For a positive fixed distance from an infinite line, the complete locus is two parallel infinite lines.
For a finite segment, use two parallel offset segments joined by semicircles centred at its endpoints: the stadium boundary.
A locus at an exact distance is the boundary, not the whole region within that distance.
Loci: Equidistant from Two Points or Two Lines
The locus of points equidistant from two points is the perpendicular bisector of the segment joining them.
The locus of points equidistant from two intersecting lines is the angle bisector(s) of the angles formed.
For two parallel lines, the locus is the line halfway between them.
When solving problems, identify which locus condition applies and shade the correct region.
Bearings
A bearing is an angle measured clockwise from north and written as three figures (e.g., 047°, 180°, 315°).
Always measure from the north line at the point you are measuring from.
To find the bearing of B from A, draw a north line at A and measure the angle clockwise to the line AB.
If you know the bearing of B from A, the bearing of A from B is found by adding or subtracting 180° (if the result is ≥360°, subtract 360°).
Bearing of Q from P
Common Mistakes to Avoid
Never erase construction arcs – they are part of the method and earn marks.
Match the required paired arc radii. In an angle-bisector construction, the first vertex arc may use a different radius from the subsequent matched pair.
For bearings, do not measure anticlockwise and do not use the wrong reference (e.g., south or west).
Always include leading zeros in bearings (e.g., 065°, not 65°).
When shading a locus region, check which side satisfies the condition – shade the correct side.
Use a compass construction where required, not a protractor.
Slide
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Xem trước miễn phí — 8 trên 61 câu hỏi. Đăng ký để xem tất cả.
1.Which of the following is the correct definition of a locus?
Easy
AThe path traced by a point moving according to a given rule
BA fixed point in space
CA line segment joining two points
DAn angle formed by two intersecting lines
2.In a three-figure bearing, the angle is measured clockwise from north.
Easy
True or false?
3.Which of the following is the correct construction for the perpendicular bisector of a line segment AB?
Easy
ADraw a circle with centre A and radius AB, and a circle with centre B and radius AB; join the intersection points.
BDraw a circle with centre A and radius AB, and a circle with centre B and radius half of AB; join the intersection points.
CDraw a line from A to B, then use a protractor to draw a 90° angle at the midpoint.
DDraw two arcs with centre A and B with different radii; join the intersection points.
4.A point P moves so that it is always 3 cm from a fixed point O. What is the locus of P?
Easy
AA circle of radius 3 cm centred at O
BA line segment of length 3 cm
CA square of side 3 cm
DA pair of parallel lines 3 cm apart
5.Which of the following is the correct construction for the bisector of angle ABC?
Easy
ADraw a vertex-centred arc cutting both arms, then equal-radius intersecting arcs from those two points; join the vertex to their interior intersection.
BDraw a line from B to the midpoint of AC.
CDraw a perpendicular from B to AC.
DDraw a circle centred at B with radius BA; join its intersection with BC to A.
6.Which of the following are standard constructions that can be done using only a ruler and compasses? (Select all that apply)
Medium
APerpendicular bisector of a line segment
BAngle bisector
CConstructing a triangle given three sides (SSS)
DUse a protractor instead of compass arcs when ruler-and-compasses only are specified
EConstructing a 60° angle
7.Match each locus condition to its shape.
Medium
Fixed distance from a point
Fixed distance from a line
Equidistant from two points
Equidistant from two intersecting lines
Circle
Two parallel lines
Perpendicular bisector
Two angle bisectors
8.Arrange the steps to construct the perpendicular bisector of a line segment AB in the correct order.
Medium
Open the compasses to a radius greater than half of AB.
Draw an arc with centre A, above and below AB.
Keeping the same radius, draw an arc with centre B, above and below AB.
Draw a straight line through the two intersection points of the arcs.
Thêm câu hỏi
9.A ship sails from port P on a bearing of 065° for 50 km to point Q. What is the bearing of P from Q?
Medium
A245°
B295°
C115°
D065°
10.When constructing a perpendicular bisector, you must leave the construction arcs visible in the final diagram.
Medium
True or false?
11.A point P is equidistant from two fixed points A and B. Which of the following is the locus of P?
Medium
AThe perpendicular bisector of AB
BThe angle bisector of angle APB
CA circle with diameter AB
DA line parallel to AB
12.A goat is tied outside a corner of an impassable rectangular shed measuring 6 m by 8 m, with a 5 m rope. The rope cannot reach either neighbouring corner. Which shape describes the grazing region?
Hard
AA circle of radius 5 m
BA circle of radius 5 m with a quarter cut out
CA circle of radius 5 m with a rectangle cut out
DA semicircle of radius 5 m
13.What is the locus of points 3 cm from A?
Medium
AA circle of radius 3 cm centred at A
BA circle of radius 6 cm centred at A
CA line perpendicular to AB
DA pair of parallel lines 3 cm from A
14.What is the locus of points equidistant from A and B?
Medium
AThe perpendicular bisector of AB
BThe angle bisector of angle APB
CA circle with diameter AB
DA line parallel to AB
15.In the construction of an angle bisector, which of the following mistakes would cause the construction to be incorrect?
Hard
AUsing different radii for the two arcs drawn from the arm-crossing points
BLeaving the construction arcs visible
CUsing a sharp pencil
DDrawing the bisector line through the vertex
16.When measuring a bearing, you should measure anticlockwise from south.
Medium
True or false?
17.Arrange the steps to construct an angle bisector of angle ABC in the correct order.
Medium
With the compass point at B, draw an arc that cuts both arms of the angle, creating points D and E.
With the compass point at D, draw an arc inside the angle.
Keeping the same radius, with the compass point at E, draw another arc inside the angle, intersecting the previous arc at F.
Draw a straight line from B through F.
18.A and B are trees 12 m apart. Rock R is the midpoint of AB. Treasure is equidistant from A and B and exactly 10 m from R. How many positions are possible?
Hard
A0
B1
C2
D4
19.In a triangle construction from SSS, the order of drawing the sides does not matter.
Easy
True or false?
20.A point P moves such that its distance from a fixed line l is always 2 cm. Which of the following is the locus of P?
Hard
ATwo parallel lines, each 2 cm from l
BA circle of radius 2 cm
CA single line parallel to l at distance 2 cm
DA pair of perpendicular lines
21.Match each locus condition to the correct shape.
Medium
Fixed distance from a point
Fixed distance from a line
Equidistant from two points
Circle
Two parallel lines
Perpendicular bisector
22.Arrange the steps to construct the perpendicular from a point P to a line l using a ruler and compasses in the correct order.
Medium
Draw an arc with centre P to cut l at two points A and B.
Draw two arcs with the same radius, centres A and B, intersecting below l.
Join P to the intersection of the arcs with a straight line.
Label the intersection of the arcs as Q.
23.Which of the following are true about three-figure bearings? (Select all that apply)
Medium
AThey are measured clockwise from north.
BThey are always written with three digits, e.g. 065°.
CThey can be measured anticlockwise from south.
DA bearing of 000° points north.
EA bearing of 090° points west.
24.The bearing of B from A is 130°. What is the bearing of A from B?
Medium
A230°
B310°
C050°
D130°
25.When constructing an angle bisector, you must use the same compass radius for both arcs drawn from the points where the first arc crosses the arms of the angle.
Easy
True or false?
26.A ship sails from port P on a bearing of 120° for 30 km to point Q. What is the bearing of P from Q?
Medium
A300°
B240°
C060°
D120°
27.Which of the following are correct statements about constructing triangles from given information? (Select all that apply)
Medium
ASSS: draw the base, then use compasses to find the third vertex.
BSAS: the angle must be between the two given sides.
CASA: the side must be between the two given angles.
DSSA always gives a unique triangle.
EConstruction arcs should be erased after completing the triangle.
28.A point Q moves so that it is always 3 cm from a fixed line l. Which of the following best describes the locus of Q?
Medium
AA circle of radius 3 cm
BTwo parallel lines, each 3 cm from l
CA single line parallel to l at distance 3 cm
DA pair of perpendicular lines
29.When using a compass to construct a perpendicular bisector, you must keep the compass radius the same for both arcs drawn from the two endpoints.
Easy
True or false?
30.A point R moves so that it is equidistant from two fixed points A and B. Which of the following is the locus of R?
Medium
AA circle with centre at the midpoint of AB
BThe perpendicular bisector of AB
CA line parallel to AB
DAn angle bisector of the angle formed by AB
31.Which of the following is the correct construction for the perpendicular from a point P to a line l?
Medium
AJoin P to an arbitrary point on l without checking the angle
BDraw an arc centred at P to cut l at two points, then construct the perpendicular bisector of the segment between those points
CDraw a line from P to any point on l and bisect it
DDraw a circle centred at P with radius equal to the distance to l
32.Which of the following are essential steps when constructing an angle bisector with a compass and ruler? (Select all that apply)
Medium
ADraw an arc centred at the vertex that crosses both arms of the angle
BFrom each intersection point, draw arcs of equal radius that intersect
CDraw a line from the vertex through the intersection of the two arcs
DMeasure the angle with a protractor to find the midpoint
EExtend the arms of the angle before bisecting
33.When a ruler and protractor are allowed, arrange the steps for an accurate SAS triangle drawing.
Medium
Draw a line segment for one side
At one endpoint, construct the given angle using a protractor
Mark off the second side length along the angle ray
Join the endpoint of the second side to the other end of the first side
34.Match each locus condition to the correct geometric shape.
Medium
Fixed distance from a point
Fixed distance from a line
Equidistant from two points
Equidistant from two intersecting lines
Circle
Two parallel lines
Perpendicular bisector
Angle bisectors
35.A point P moves so that it is always equidistant from two fixed points A and B. Which of the following is the locus of P?
Medium
AThe perpendicular bisector of AB
BA circle with centre A
CA circle with centre B
DThe line AB
36.Which of the following are correct statements about three-figure bearings? (Select all that apply)
Medium
ABearings are measured clockwise from north.
BBearings are always written with three digits, including leading zeros.
C065° and 65° have the same angular magnitude, but only 065° uses three-figure notation.
DThe bearing of a point directly east is 090°.
EBearings can be measured anticlockwise from south.
37.When constructing a triangle from SSS, the order in which you draw the sides does not affect the final triangle.
Medium
True or false?
38.Which direction is a three-figure bearing measured from?
Easy
ANorth, clockwise
BNorth, anticlockwise
CSouth, clockwise
DSouth, anticlockwise
39.When constructing a perpendicular bisector, you must use the same compass radius for both arcs drawn from the two endpoints.
Easy
True or false?
40.A point P moves so that it is always 3 cm from a fixed line l. Which of the following best describes the locus of P?
Easy
AA circle of radius 3 cm
BTwo parallel lines, each 3 cm from l
CA single line parallel to l at distance 3 cm
DA rectangle of width 3 cm
41.A three-figure bearing is written 065°. What is its numerical angle in degrees?
Easy
42.Complete the sentence.
Easy
A bearing is measured ____ from north.
43.What is the bearing of a point that is directly west of the observer? Give your answer as a three-figure bearing.
Easy
44.The bearing of B from A is 120°. What is the bearing of A from B?
Medium
45.A point moves so that it is always the same distance from two intersecting lines. What is the locus?
Medium
46.A and B are 6 cm apart. How many points are exactly 3 cm from A and equidistant from A and B?
Hard
47.The bearing of P from Q is 250°. What is the bearing of Q from P?
Medium
48.State the bearing that is exactly opposite to a bearing of 135°.
Medium
49.A three-figure bearing is written 045°. What is its numerical angle in degrees?
Easy
50.Complete the sentence about the locus of a point moving at a fixed distance from a line.
Medium
The locus of a point that is always a fixed distance from a given line is a pair of ____ lines, one on each side of the given line.
51.A point P moves so that it is always 4 cm from a line l. Describe the complete locus of P.
Medium
52.A point X is 5 cm from a fixed point O. State the length of the radius of the circle that is the locus of X.
Medium
53.Describe how you would construct the perpendicular from a point P to a line l using only a ruler and compasses.
Hard
54.Describe how you would construct the perpendicular bisector of a line segment AB using only a ruler and compasses.
Hard
55.An impassable rectangular fenced field is 60 m by 40 m. A goat is tied outside one corner by a 30 m rope and cannot cross the fence. Describe its grazing region outside the field and find its area in terms of π.
Hard
56.Write the bearing 65° as a three-figure bearing.
Easy
A006°
B056°
C065°
D105°
57.The bearing of B from A is 120°. What is the bearing of A from B?
Medium
A360 °
B300 °
C330 °
D270 °
58.A and B are 6 cm apart. How many points are exactly 3 cm from A and equidistant from A and B?
Hard
A1
B3
C2
D4
59.The bearing of P from Q is 250°. What is the bearing of Q from P?
Medium
A63 °
B70 °
C84 °
D77 °
60.Convert the bearing 45° to a three-figure bearing.
Easy
A004°
B040°
C054°
D045°
61.A point X is 5 cm from a fixed point O. State the length of the radius of the circle that is the locus of X.
Medium
A7 cm
B5 cm
C4 cm
D6 cm
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