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Geometry and Measure
GCSE Grade G
Reading Scales
Time Measure
There are two things to do when reading a scale.
There are 60 seconds in 1 minute
1.
2.
Make sure you know what each division on the scale represents.
Make sure you read in the right direction.
There are 60 minutes in 1 hour
There are 24 hours in 1 day
There are 30 days in April, June, September and November
There are 31 days in January, March, May, July, August, October and December.
There are 28 days in February if the year is not a leap year.
There are 29 days in February if the year is a leap year.
There are 7 days in a week
There are 52 weeks in a year
There are 12 months in 1 year.
Time Measures
Measure
When adding or subtracting time, it is best to add or subtract the hours and minutes separately.
Length – metres (m), kilometres (km), centimetres (cm)
Example
Weight – gram (g), kilograms (kg), tonne
On Saturday, Dale practised the guitar from 11.40 a.m. until 1 p.m. and then from 4.25 p.m. until 5.15 p.m.
Volume/Capacity – litre (l), centilitre (cl), millilitre (ml)
For how long did Dale practise on Saturday?
Time – years, weeks, days, hours and minutes
Answer
Time
From 11.40 a.m. until 1 p.m. is 1 hour 20 minutes.
This clock has two hands
From 44.25 p.m. until 5.15 p.m. is 35 minutes + 15 minutes.
The short hand tells us the hour and the long hand tells us the minutes.
Total hours = 1 hour
For the hour hand each mark is one hour.
For the minute hand each mark is 5 minutes.
Total minutes = 20 + 35 + 15
= 70 minutes
= 1 hour 10 minutes.
This clock reads 20 past 8. We write this as 8:20.
Total time = 1 hour + 1 hour 10 minutes = 2 hours 10 minutes.
Times before noon have a.m. beside them
Types of Angle
Times after noon have p.m.
A right angle is a ¼ turn.
7:45 at night is 7:45 p.m.
A straight angle is a ½ turn.
8:20 in the morning is 8:20 a.m.
An acute angle is smaller than a right angle.
24 Hour Time
An obtuse angle is greater than a right angle but smaller than a straight angle.
Twenty past eight could be in the morning or at night
A reflex angle is greater than a straight angle but less than a complete turn.
The 24 hour clock numbers the hours after midday as 13, 14, 15,
There are 90° in a ¼ turn.
A turn of 45° is ½ of a ¼ turn
0000 0100 0200 0300 0400 0500 0600 0700 0800 0900 1000 1100 1200
12am 1am 2am 3am 4am 5am 6am 7am 8am 9am 10am 11am 12pm
midnight
noon
Afternoon and Evening
1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 0000
12pm 1pm 2pm 3pm 4pm 5pm 6pm 7pm 8pm 9pm 10pm 11pm 12pm
noon
midnight
3:20 pm is written as 1520 hours on a 24 hour clock.
A turn of 30° is 1/3 of a ¼ turn.
A turn of 60° is 2/3 of a ¼ turn
Perpendicular and Parallel Lines
D
A
Parallel lines are always the same distance apart.
C
Parallel lines never meet.
In this diagram the line AB is parallel to the line CD.
We write AB//CD. The symbol // is read as “is parallel to”
We often put arrows on lines to show they are parallel.
We say this as “fifteen twenty hours”
Perpendicular lines are at right angles to each other.
In the diagram the line AB is perpendicular to the line CD.
Twenty four hour clock times always have 4 digits.
We write AB ⊥ CD. The symbol ⊥ is read as “is perpendicular to”⊥
8:20 am is written as 0820.
B
This line is called AB since it runs from A to B.
It can also be called BA since it runs from B to A.
C
A
B
D
Perimeter
Naming Polygons
The distance around the outside of a shape is called the perimeter
A 3-sided polygon is a triangle
A 5-sided polygon is a pentagon
A 7-sided polygon is a heptagon
A 9-sided polygon is a nonagon
Perimeter is measured in mm, cm, m, or km.
Example
Find the perimeter of this rectangle.
7 cm
A 4 sided polygon is a quadrilateral
A 6 sided polygon is a hexagon
An 8 sided polygon is a octagon
A 10 sided polygon is a decagon
A regular polygon has all its sides equal and all its angles equal.
Vertex
Edge
Faces Edges Vertices
A face is a flat surface.
This shape has 6 faces.
3 cm
An edge is a line where two faces meet.
This shape has 12 edges.
Answer
A vertex is a corner where edges meet (Vertices is the plural of vertex).
This shape has 8 vertices.
Perimeter = 7 + 3 + 7 + 3 = 20 cm
For Grade F the perimeter of a compound shape would have to be found.
Face
Diagonal Lines
2cm
The lines drawn from one corner to
the opposite corner are called diagonals
Example
Find the perimeter of this compound shape.
Circles
Answer
A circle is drawn accurately using a compass.
8cm
The lengths of the two missing sides are
6 cm and 4 cm.
The line from the centre of the circle to the circumference is called the radius
The diameter of the circle is the width.
So the perimeter = 2 + 6 + 4 + 2 + 6 + 8 = 28 cm
2cm
Circumference
6cm
Area
Radius
The amount of surface a shape covers is called its area.
This square is 1cm long and 1cm wide.
The area of this square is called 1 square centimetre.
We write 1 square centimetre as 1cm2
Diameter
Some units for measuring area are mm2, cm2, m2, km2.
Volume
Congruence
The volume of a shape is the amount of space it takes up
Identical shapes are known as congruent shapes.
1 cubic centimetre is the amount of space this cube takes up.
1 cubic centimetre is written 1cm3. We say that the volume of
a cube of side 1cm is 1cm3.
If a tracing of one shape will fit exactly over another shape, the shapes are said to be congruent
Congruent shapes are the same shape and size.
These shapes are congruent
Units of volume are mm3, cm3, m3
Congruent shapes:
Special Triangles and Quadrilaterals
o
are the same size;
o
have angles of the same size;
o
have sides of the same length;
Congruent solids have angles and sides and faces of the same size.
Equilateral
Isosceles
Right-angled
Scalene
An equilateral triangle has all its sides equal and all its angles equal.
An isosceles triangle has two equal sides and two equal angles. The equal angles are opposite the equal
sides.
A right-angled triangle has one angle equal to 90°.
A scalene triangle has not equal sides and no equal angles.
GCSE Grade F
Estimating
Step 4
Sometimes when we estimate it is better to overestimate and sometimes it is better to underestimate
Decide whether to use the inside or outside scale. The scale to use is the one that has 0° on the
arm of the angle.
Overestimate – amount of paint for a room
Step 5
Read off the number where the other arm of the angle meets the chosen scale.
Underestimate – number of people in a lift.
Step 6
Check this number with your estimate to make sure you have read the correct scale.
If we know the height of something we can estimate the height or length of other objects.
The size of an angle which is greater than half a circle can be read directly
from a circular protractor. If you have a semicircular protractor, then take the following steps.
Metric Units
Length Conversions
For converting between the commonly used units for length the relationships in this list can be used
Step 1
Measure the inside angle.
Step 2
Subtract this measurement from 360°
1km = 1000m
1m = 1000mm
1m = 100cm
1cm = 10mm
To change small units to larger units, always divide.
Example
732 cm to metres
Answer
732 ÷ 100 = 7.32 m
To change large units to smaller units, always multiply.
Example
Change 1.2 m to centimetres Answer
1.2 x 100 = 120 cm
Drawing Angles
Step 1
Draw a straight line. This will be one of the arms of the angle.
Step 2
Place the protractor so that the base line lies along the drawn line and
the centre of the protractor is on the end of the drawn line.
Step 3
Read around the scale that begins 0° on the drawn line. Put a small mark beside 130°
(for example).
To change large units to smaller units, always multiply.
Step 4
Take the protractor away. Through the small mark, draw the other arm of the angle.
Capacity Conversions
100cl = 1000ml
Angles of any size between 0° and 180° are as above.
Using a circular protractor, angles of any size may be drawn.
Using a semicircular protractor, angles between 180° and 360° are as follows.
To change small units to larger units, always divide.
Step 1
Subtract the size of the angle to be drawn from 360°
To change large units to smaller units, always multiply.
Step 2
Draw this smaller angle. The other angle in the diagram will now be the angle required.
Volume
Naming Angles
Weight Conversions
1kg = 1000g
1tonne = 1000kg
To change small units to larger units, always divide.
1cl = 10ml
1000 litres = 1 m3
1 ml = 1 cm
3
Angles
Estimating Angle Size
The best estimate you would be expected to give would be to the nearest 10°.
When estimating the size of an angle it is often helpful to compare with a right angle.
Measuring Angles
We measure the size of an angle with a protractor.
There are two sorts of protractor; a circular one and a semicircular one.
Follow these steps to measure an angle using the protractor.
An angle can be named by the capital letter at the vertex.
The marked angle can be named as the angle Q or as ∠Q.
The symbol ∠ is the symbol for angle.
R
An angle can also be named by using three capital letters.
The marked angle could be named as ∠PQR or ∠RQP.
When we use three capital letters to name an angle,
the first is the letter at the end of one of the arms, the middle
letter is the letter at the vertex and the last letter is the letter
at the end of the other arm.
Step 1
Estimate the size of the angle.
Step 2
Place the centre of the protractor on the vertex of the angle.
We cannot name the marked angle as ∠Q since there is more
than one angle at Q. There is ∠PQS and ∠RQP taking these
two angles together there is RQS. Whenever there is more
than one angle at a point we must use three letters to name
the angle we want.
Step 3
Keeping the centre on the vertex, move the protractor around until the base line lies along one of
the arms of the angle.
Always remember that when an angle is named by three capital
letters, it is the middle letter that is the vertex.
Q
P
R
Q
P
S
Angle Facts
The angles on a straight line add up to 180°
a
Angles such as a, b, c and d are
called angles at a point.
They are the angles around a point.
b
(called the scale).
Example: this drawing has a scale of 1:10, so anything drawn with
the size of 1 would have a size of 10 in the real world,
so a measurement of 150mm on the drawing would be 1500mm on the real horse.
b
c
d
a
Angles at a point add up 360°
Lines of Symmetry
A line of symmetry is a line that can be drawn through a shape so that what can be seen on one side of the
line is the mirror image of what is on the other side. This is why a line of symmetry is sometimes called a
mirror line. It is also the line along which a shape can be folded exactly.
Real Horse
Drawn Horse
1500 mm high
150mm high
Nets
A net is a flat shape that can be folded into a 3D shape.
The Net
Example
Completed Box
Find the number of lines of symmetry for this cross.
Answer
This cross has a total of
four lines of symmetry.
Area of a Rectangle
Area = length x width
or
Area = base x height
width
length
Rotational Symmetry
The order of rotational symmetry is the number of times a shape fits exactly onto itself during one complete
turn
Example
Since all shapes will fit onto themselves at least once during a complete turn all shapes have order of
rotational symmetry of at least 1
Answer
A 2D shape has rotational symmetry if it can be rotated about a point to look exactly the same in a new
position.
height
base
Calculate the area of this rectangle.
4 cm
Area of rectangle = length x width
= 11 cm x 4 cm
= 44 cm2
11 cm
The easiest way to find the order of rotational symmetry for any shape is to trace it and count the number of
times that the shape stays the same as you turn the tracing paper through one complete turn.
For Grade D the area of a compound shape would have to be found.
Example
Example
Find the order of rotational symmetry for this shape.
Find the area of this compound shape.
First hold the tracing paper on top of the shape and trace the shape.
Then rotate the tracing paper and count the number of times the tracing
paper matches the original shape in one complete turn.
Answer
You will find three different positions.
Then calculate the area of each one.
So, the order of rotational symmetry for the shape is 3.
Scale Drawings
A scale drawing shows a real object with accurate sizes except
they have all been reduced or enlarged by a certain amount
2cm
2cm
8cm
First split the shape into two rectangles, A and B.
8cm
area of A = 2 x 6 = 12 cm2
2cm
area of B = 6 x 2 = 12 cm2
6cm
The area of the shape is given by:
area of A + area of B = 12 + 12 = 24 cm
2
A
B
6cm
2cm
GCSE Grade E
1 gallon is about 4.5 litres.
40 gallons is about 40 x 4.5 litres or 180 litres.
That is, the tank holds about 180 litres of water.
Imperial Units
Imperial Measures for Length, Weight and Capacity
Length – inch (“), foot (‘), yard, mile
Length Equivalents
Mass – ounce (oz), pound (lb), stone, ton
Capacity – pint, gallon
Length
Capacity
1 foot = 12 inches
Weight
1 lb = 16 oz
1 yard = 3 feet
1 stone = 14 pounds
1 mile = 1760 yards
1 ton = 2240 pounds
1 gallon = 8 pints
To change small units to larger units, always divide.
To change large units to smaller units, always multiply.
Example
Write 5’4” in inches
Answer
5 feet
= 12 x 5 inches
= 60 inches
Then 5’4”
= 60 inches + 4 inches
= 64 inches
5 mile is about 8km
3 feet is a little less than 1 m
1 inch is about 2.5 cm
Example
Dee walked 20km. About how many miles did Dee walk?
Answer
Firstly, find how many lots of 8km Dee walked.
We do this by dividing 20 by 8. We get 2.5
Since there are 5 miles in each lot of 8km, we now multiply by 5 to find the total number of miles.
We get 5 x 2.5 = 12.5 miles.
That is, Dee walked about 12 ½ miles.
Triangles and Angles
The sum of the interior angles of a triangle is 180°
When adding, measurements which are given in mixed units begin by adding the units separately.
Whenever we use the fact that the angles inside a triangle add to 180°,
we must say so, we may use ∠ sum of a ∆
The same applies when subtracting or multiplying.
Example
When dividing, it is wise to convert the measurement so that it is given in the smaller unit.
Find the value of n
Answer
Michael needs 2 x 2 lb 12 oz of apples.
2 x 2 lb = 4 lb
2 x 12 oz = 24 oz
= 1 lb 8 oz
So Michael needs 4 lb + 1 lb 8 oz = 5 lb 8 oz of apples.
Mass Equivalents
1kg is about 2 lb
1 lb is about 0.5kg or 500g
Example
118°
n
Example
A recipe for making apple strudel uses 2lb 12 oz of apples. Michael was using this recipe to make strudel
for his family.
He was going to double the recipe. What quantity of apples should he use?
30°
Answer
n + 30° + 118° = 180 (∠ sum of a ∆ )
n + 148° = 180°
n = 32° (subtracting 148° from both sides)
Angles in Special Triangles
An equilateral triangle has all its sides equal and all its angles equal.
Each angle is 60°
An isosceles triangle has two equal
sides and two equal angles.
A recipe for fudge cake uses 8oz of butter. About how many grams is this?
Answer
1 lb of butter is about 500g. 8oz is ½ lb.
½ lb of butter is about ½ x 500g or 250g
Capacity Equivalents
1 Gallon is about 4.5 litres
1 litre is about 1.75 pints
Example
A tank holds 40 gallons of water. About how many litres is this?
Answer
a
The angles a, b, c, are called the interior angles of a triangle.
They are the angles inside the triangle.
A right-angled triangle has an interior angle of 90o
b
c
Surface Area and Volume of a Cuboid
A cuboid is a box shape, all six faces are rectangles.
The volume of a cuboid is given by the formula:
Height (h)
Volume = length x width x height
The surface area of a cuboid is calculated by finding the
total area of the six faces, which are rectangles.
The opposite rectangles have the same area.
The area of the top and bottom are the same.
The area of the front and back are the same.
The area of the two sides rectangles are the same.
Width (w)
Length (l)
Surface area of a cuboid is given by the formula:
Surface area = 2lw + 2hw + 2hl
Example
Calculate the volume and surface area of this cuboid
4cm
Answer
V = 5 x 3 x 4 = 60cm3
S. Area
3cm
= (2 x 3 x 5) + ( 2 x 4 x 5) + ( 2 x 4 x 3)
= 30 + 40 + 24
= 94 cm2
5cm
For Grade D the volume of a compound shape would have to be found.
Tessellations – A repeating pattern of identical shapes which fit together exactly, leaving no gaps.
Reflection
Reflective symmetry can be described by giving the location of the lines of symmetry. Lines of symmetry are
often called axes of symmetry.
Example
On the grid, reflect triangle P in the y-axis.
The gray triangle is the original shape and the red triangle is the image.
y
6
5
4
AB
P
3
2
1
–6
–5
–4
–3
–2
–1
O
–1
–2
–3
–4
–5
–6
1
2
3
4
5
6
x
GCSE Grade D
N
Compass Directions
Special Quadrilaterals
N 60°E
N 60° E means 60° East of North
Parallelogram
•
•
•
•
To find this direction follow these steps.
Step 1
Face North
Step 2
Turn 60° towards East.
60°
Opposites sides are parallel
Opposite sides are equal
Diagonals bisect each other
Opposite angles are equal
Bearings
The bearing of a point B from a point A is the angle through which you turn clockwise as you change
direction from due north to the direction of B.
Rhombus
For example in the diagram above the bearing is 060°
1.
2.
3.
Always measure from North.
Always move in a clockwise direction.
Always give the answer in three figures.
b
a
They are the two angles which are opposite one another
(not beside each other) when two lines intersect.
Vertically opposite angles are equal
•
•
•
A kite is a quadrilateral with two pairs of equal adjacent sides.
Its longer diagonal bisects its shorter diagonal at right angles.
The opposite angles between the sides of different lengths are equal.
Trapezium
a
Angles such as a and b are called corresponding angles.
Corresponding angles are in corresponding positions.
One corresponding angles can be translated onto the other.
They are sometimes called F angles.
•
a=b
b
A trapezium has two parallel sides.
Using Isometric Paper to Draw 3-D Shapes
Isometric means “same measure”
On an isometric drawing of a 3-D shape the lengths which are equal on the shape are also equal on the
drawing.
Corresponding angles are equal
a=b
Angles such as a and b are called alternate angles.
Alternate angles are both inside the parallel
lines and on opposite sides of the transversal.
One alternate angle can be rotated onto
another alternate angle. They are sometimes called Z angles.
A rhombus is a parallelogram which has all sides equal.
Its diagonals bisect each other at right angles.
Its diagonals also bisect the angles
Kite
Intersecting and Parallel Lines
Angles sub as a and b are called vertically opposite angles.
•
•
•
An isometric drawing of a 3-D shape will make the shape appear slightly distorted.
To make an isometric drawing of a 3-D shape we can use either isometric graph paper or isometric dot
paper.
a
Vertical edges of the 3-D shape should be drawn as vertical lines.
b
Three steps to drawing a cube using isometric paper
Alternate angles are equal
Angles such as a and b are called interior angles.
Interior angles are both inside the parallel
lines and on the same side of the transversal.
a
a + b = 180°
Interior angles add to 180°
There is often more than one way of finding an unknown angle. We can use any of

vertically opposite angles

adjacent angles on a straight line

angles at a point

corresponding angles

alternate angles

interior angles
b
Below are two drawings of the same cuboid on different isometric grids
Area of a Triangle
Translations
height
height
Area = base x height
2
A translation is the movement of a shape from one position to another without reflecting it or rotating it. It is
sometime called a ‘sliding’ transformation, since the shape appears to slide from one position to another.
base
Example
base
Example
Describe the translation of the red arrow to the grey arrow.
Answer
Find the area of this right-angled triangle.
4 cm
Answer
The arrow has been translated 5 squares right and
1 square down.
Area = ½ x 7 x 4 = 14 cm2
b
A translation can also be described by using a vector.
(Grade C)
𝑎
A vector is written in the form � � where a describes the
𝑏
horizontal movement (x-direction) and b describes the
vertical movement (y-direction)
6 cm
Giving the answer in vector form for the example above is �
7 cm
Area of Parallelogram
h
Area of a parallelogram = base x height
A=bxh
Example
Find the area of this parallelogram
Answer
Rotation
Area = 8 x 6 = 48 cm2
Area of a trapezium = half the sum of the parallel sides x height,
h
the height is the perpendicular distance between the parallel sides.
A = ½ (a + b) x h
4 cm
Example
Find the area of the trapezium
b
Area = ½ (4 + 7) x 3 = 16.5 cm2
7 cm
Interior Angles of a Polygon
The object (grey shape) has been rotated 900 (or ¼ turn)
clockwise to give the image (red shape)
To find the sum of the angles in a polygon we can proceed as follows
Step 1
From one vertex, draw all the diagonals to divide the polygon into triangles.
Step 2
Find the sum of the angles in all of these triangles.
Object
Image
Enlargement
Scale Factor of an enlargement
A’
If a shape is enlarged so that each length
becomes twice the size it was on the original
we say the scale factor of the enlargement is 2.
3 cm
Answer
A rotation transforms a 2D shape to a new position
by turning it bout a fixed point, called the centre of
rotation.
a
8 cm
Area of Trapezium
5
�.
−1
Shape ABCD enlarged to the shape A′ B′ C′ D′
Each length on the enlargement A′ B′ C′ D′ is
twice as long as the corresponding length on
the original ABCD.
The scale factor for this enlargement is 2
Example
B’
B
A
C’
D’
D
C
Find the sum of the angles in this polygon.
The triangle ABC has been enlarged to the
triangle A′B′C′.
The scale factor of the enlargement is 3.
If you join A’A , B’B and C’C, the point P,
where these lines meet is known as the
centre of enlargement.
Answer
This polygon can be divided into 4 triangles.
Sum of the angles in 4 triangles = 4 x 180° = 720°
4
1
2
3
117
Calculate a Missing Angle in a Polygon
o
98o
Example
a
Work out the size of the angle marked a.
Answer
0
The sum of the angles in a hexagon total 720 , (see above).
0
0
0
0
0
0
134o
118o
Total the angles given and subtract from 7200.
0
720 – (117 + 98 + 134 + 104 + 118 ) = 149 .
104
o
Drawing Enlargements
To draw an enlargement of a shape we need
to know both the scale factor and the centre
of enlargement.
We need the scale factor to make the
enlargement the right size.
We need the centre of enlargement to position
the enlargement correctly.
A′
A
P
C
B
C′
B′
GCSE Grade C
Circumference and Area of a Circle
Exterior Angles of a Polygon
C = πd
The exterior angles of any sided polygon total 360°.
We can use the formula C = πd to calculate the circumference of a circle
Remember
r is the radius of a circle.
d = 2r
e
i
Replacing d by 2r in the formula C = πd we get C = 2πr
Example
e i
Find the circumference of these circles
a.
i e
i
e
b.
45mm
Example
4 cm
The diagram shows part of a regular octagon.
x
Work out the size of angle x.
Diagram NOT accurately drawn
Answer
Knowing that the exterior angles sum to 360°,
360 ÷ 8 = 45
Answer
a.
C = πd
And knowing Adjacent angles on a straight line add to 180°
C = 2 x 3.1 x 45
C = 3.1 x 4
180 – 45 = 135
C = 279mm
C = 12.4cm
C = 2πr
b.
x = 135°
The area of a circle can be calculated using the formula A = πr2.
Pythagoras’ Theorem
Example
Pythagoras’ Theorem gives the relationship between
the lengths of the sides in a right-angled triangle.
Find the area of these circles
2
2
c
2
c =a +b
Take π = 3.1
is Pythagoras’ Theorem in symbols for the triangle shown
Give the answers to 1 d.p.
a.
b.
We can use Pythagoras’ Theorem to find the length of the third side of a right-angled triangle, if we
know the lengths of the other two sides.
Example
Find the value of x in each of the triangles.
Answer
a.
r = 39.5mm
2
A = 3.1 x 39.5
A = 4836.8mm2 (1d.p.)
Answer
Prisms
a.
A prism is a 3D shape that has the same cross-section running all the way through it, whenever it is
cut perpendicular to its length.
Cross-section
isosceles triangle
x
15cm
9cm
2
A = 57.3 cm (1d.p.)d = 79mm
Triangular prism
b.
7cm
A = πr2
A = 3.1 x 4.32
Cuboid
Cross-section
rectangle
a
Using Pythagoras’ Theorem
79 mm
4.3cm
A = πr2
b
Cylinder
Cross-section
circle
Hexagonal prism
Cross-section
regular hexagon
The volume of a prism is found by multiplying the area of its cross-section by the length of the
prism.
9cm
x
x2 = 72 + 92
(Pythagoras’ Theorem)
= 49 + 81
= 130
x = √130
= 11.4cm (to 1d.p.)
b.
152 = x2 + 92
Rewrite with x2 first
x2 + 92 = 152
x2 = 152 – 92
= 225 – 81
= 144
x = √144
= 12
(Pythagoras’ Theorem)
(subtracting 92 from both sides)
Locus
The locus of an object is the set of all the possible positions that this object can occupy.
In particular the path of an object, moving according to some rule, is the locus of the object.
(The plural of locus is loci) sometimes the locus can be described in words; sometimes
it can be better described by a sketch.
To find the locus of a moving point, always sketch a few possible positions of the point.
locus
1. The locus of a point
which is a constant
distance from a fixed
point is a circle
A
2. The locus of a point
locus
which is a constant
distance from a fixed
line is a pair of parallel lines
3. The locus of a point
which is equidistant from
two fixed points is the
perpendicular bisector of the line
joining the fixed points
4. The locus of a point
which is equidistant from two
intersecting lines
is the pair of lines
which bisect the angles
between the fixed lines
locus
A
B
locus
locus