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79
Types of quadrilateral
Name Shape Properties
1. Quadrilateral ▪ 4 sides (quad.) ▪ sum of interior 's 360
2. Kite ▪ quadrilateral▪ 2 pairs of equal adjacent sides▪ a pair of equal opposite 's▪ a diagonal bisects angles▪ a diagonal bisected at 90º
3. Trapezium ▪ quadrilateral (trap.) ▪ a pair of parallel sides
4. Parallelogram ▪ trapezium (parm) ▪ 2 pairs of parallel sides
▪ opposite sides equal▪ opposite angles equal▪ diagonals bisect each other
5. Rectangle ▪ parallelogram (rect.) ▪ a right angle ▪ all interior 's 90
▪ diagonals equal
6. Rhombus ▪ parallellogram (rhom.) ▪ kite
▪ all sides equal▪ diagonals perpendicular▪ diagonals bisect angles
7. Square ▪ rectangle (with all sides equal)▪ rhombus (with all
interior 's 90 )
80
SQUARE
RHOMBUS
RECTANGLE
PARALLELOGRAM
TRAPEZIUM
KITE
QUADRILATERAL O O O O O P O O O O O O O O O O
All
side
s eq
ual
Two
pairs
of a
djac
ent
side
s eq
ual
Opp
osite
side
s eq
ual
A p
air
of s
ides
par
allel
Opp
osite
side
s pa
ralle
l
Inte
rior
ang
le s
um 3
60º
All
angles
90º
A p
air
of e
qual o
ppos
ite
angles
Both
pairs
of
oppo
site
an
gles
equ
alDiago
nals b
isec
t ea
ch
othe
ra
diag
onal b
isec
ted
at
90º
Both
diago
nals b
isec
ted
at 9
0º
Diago
nals e
qual
Diago
nal bi
sect
s a
pair
of o
ppos
ite
angles
Diago
nal bi
sect
s bo
th
pairs
of o
ppos
ite
angles
All
angles
bet
wee
n diag
onal a
nd s
ides
45º
4 sidesa pair of// sides 2 pairs of
// sides
parallelogramwith a
right angle
parallelogramwith a pair of = adjacent sides
2 pairs of = adjacent sides
rhombus with a right angle /
rectanglewith a pair of = adjacent sides
The diagram below illustrate the relationship between the quadrilaterals
§ Exercise 1 Complete the table:
»
81
120º a
80º
110º
a
140º
50º 30º
45º ab
c
a
b c d
22º
80º
58º
25
75a
b
60
30
x
y
8
5 2y
1x
3
4
x
P A
RM
75°4x
2x3x
1
R H
B
74°
2
O
M2
11
2
12
§ Exercise 2
Find the sizes of the angles marked with letters:
2.1 2.2
2.3 2.4
Find the unknown lengths:
2.5 2.6
2.7 2.8
2.9 Find the values of the x and angle ARM.
2.10 RHMB is a rhombus. Calculate, giving reasons:(a) BMH
(b) 2H
82
A R
PC
20°
64°
35°
R
O
T
1
1
2
2
112°
E
C
R(6;2)
P(-7;-2)
Q(2;4)
x
y
S(-6;-6)
R(5;-2)
P(-8;2)
Q(2;4)
x
y
S(-5;-4)
2.11 Calculate A,
giving reasons: 2.12 Determine the size of 1T
in rectangle RECT.
How do we prove that a quadrilateral is a trapezium?
By showing that it has a pair of parallel sides.
**e.g.1 Show that PQRS is a trapezium.
PQ SRi.e. ... that PQ / /SR ... m m
Answer:
P QPQ
P Q
2 4 2
7 2 3
y ym
x x
S RSR PQ
S R
6 2 2
6 6 3
y ym m
x x
PQ//SR PQRS trapezium
How do we prove that a quadrilateral is a parallelogram?
**e.g.2 Show that PQRS is a parallelogram, using 4 different methods.
Answer:
Method 1: P QPQ
P Q
2 4 1
8 2 5
y ym
x x
S RSR PQ
S R
4 2 1
5 5 5
y ym m
x x
PQ//SR
P SPS
P S
2 42
8 5
y ym
x x
Q RQR PS
Q R
4 22
2 5
y ym m
x x
PS//QRPQRS is a parallelogram (2 pairs of // sides)
that it has two pairs of parallel sides One pair is insufficient!or
that it has two pairs of equal sides One pair is insufficient!or
that it has one pair of sides equal and parallelor
that its diagonals bisect (i.e. diagonals have the same midpoint) Usually bestor
that it has two pairs of equal opposite angles Not recommended
By showing
83
R(5;-2)
P(-8;2)
Q(2;4)
x
y
S(-5;-4)
Method 2:
2 2
P Q P Q
2 2
PQ
8 2 2 4
104
x x y y
2 2
S R S R
2 2
SR
5 5 4 2
104
x x y y
PQ SR
2 2
P S P S
2 2
PS
8 5 2 4
45
x x y y
2 2
Q R Q R
2 2
QR
2 5 4 2
45
x x y y
PS QR
PQRS is a parallelogram (opposite sides )
Method 3: P QPQ
P Q
2 4 1
8 2 5
y ym
x x
S RSR PQ
S R
4 2 1
5 5 5
y ym m
x x
PQ//SR
2 2
P Q P Q
2 2
PQ
8 2 2 4
104
x x y y
2 2
S R S R
2 2
SR
5 5 4 2
104
x x y y
PQ SR
PQRS is a parallelogram (one pair of sides and //)
Method 4:
P R P R
12
midpoint PR ;2 2
8 5 2 2;
2 2
1 ;0
x x y y
Q S Q S
12
midpoint QS ;2 2
2 5 4 4;
2 2
1 ;0
x x y y
midpoint PR midpoint QS
PQRS is a parallelogram (diagonals bisect each other)
If you had to decide which of these four methods to use ... ??
4 calculations 4 calculations 4 calculations 2 calculations
Yes, the preferred method for proving that a quadrilateral is a parallelogram is to prove that the diagonals bisect ... i.e. that the midpoints of the diagonals are the same.
84
C(1;-3)
A(-1;6)B(3;5)
x
y
D(-3;-2)
C(4;-2)
A(-4;6)B(3;5)
x
y
D(-3;-1)
How do we prove that a quadrilateral is a rectangle?
First show that the quadrilateral is a parallelogram
Then show
or
**e.g.3 Show that ABCD is a rectangle, using 2 different methods.
Answer:
First prove that ABCD is a parallelogram:(preferred method ... midpoints)
12
12
1 1 6 3Midpt AC ; 0;1
2 2
3 3 2 5Midpt BD ; 0;1 Midpt AC
2 2
ABCD is a parallelogram (diags bisect)
Now do the extra step to prove that it is a rectangle:
AD AB
6 2 6 5
1 3 1 38 1
2 41
AB AD (i.e. A 90 )
m m
or
2 22
2 22
2
AC 1 1 6 3
85
BD 3 3 5 2
85 AC
AC BD
ABCD is a rectangle
How do we prove that a quadrilateral is a rhombus?
First show that the quadrilateral is a parallelogram
Then show
or
**e.g.4 Show that ABCD is a rhombus, using 2 different methods.
Answer:
First prove that ABCD is a parallelogram:(preferred method ... midpoints)
4 4 6 2Midpt AC ; 0;2
2 2
3 3 1 5Midpt BD ; 0;2 Midpt AC
2 2
ABCD is a parallelogram (diags bisect)
that it has a right angle (i.e. two sides are perpendicular ... 1 2 1)m m
that its diagonals are equal
that it has a pair of equal adjacent sides
that its diagonals intersect at 90º
or Show that all four sides are equal
85
C(8;-1)A(-6;1)
B(2;7)
x
y
D(0;-7)
Now do the extra step to prove that it is a rhombus:
AC BD
6 2 5 1
4 4 3 31 1
1
AC BD
m m
or
2 22
2 22 2
AB 4 3 6 5 50
AD 4 3 6 1 50 AB
AB AD
ABCD is a rhombus
How do we prove that a quadrilateral is a square?
that the quadrilateral is a rectangleand
that the quadrilateral has a pair of equal adjacent sides
or that the quadrilateral is a rhombus
and that the quadrilateral has a right angle
**e.g.5 Show that ABCD is a square, using 2 different methods.
Answer:
First prove that ABCD is a parallelogram:(preferred method ... midpoints)
6 8 1 1Midpt AC ; 1;0
2 2
2 0 7 7Midpt BD ; 1;0 Midpt AC
2 2
ABCD is a parallelogram (diags bisect)
AC BD
1 1 7 7
6 8 2 01
AC BD
m m
or
2 22
2 22 2
AB 6 2 1 7 100
AD 6 0 1 7 100 AB
AB AD
ABCD is a rhombusNow do the extra step to prove that it is a square:
AB AD
1 7 1 7
6 2 6 01
AB AD
m m
or
2 22
2 22 2
AC 6 8 1 1 200
BD 2 0 7 7 200 AC
AC BD
ABCD is a square
Show:
86
S(2;4)
A(6;6)
T(1;2)
L(5;4)
x
y
C(7;-4)
A(-11;5)B(0;7)
x
y
D(-6;-5)
T(-7;2)
B(1;8)
A(-1;-6)
Rx
y
7
K(7;6)C(-4;8)
U(-9;-2)R(2;-4)
x
y
L(0;7)I(-12;9)
K(-10;-3) N(1;-4)
x
y
How do we prove that a quadrilateral is a kite?
that the quadrilateral has 2 pairs of equal adjacent sides
or that a diagonal is bisected
and that the diagonals intersect at right angles
**e.g.6 Show that ABCD is a kite.
Answer:
2 22
2 22 2
2 22
2 22 2
AB 11 0 5 7 125
AD 11 6 5 5 125 AB
BC 0 7 7 4 170
CD 7 6 4 5 170 BC
AB AD and BC CD
ABCD is a kite
§ Exercise 3
3.1 Given quadrilateral SALT with vertices S(2;4), A(6;6), L(5;4)and S(1;2), prove, in three different ways, that SALT is a parallelogram.
3.2 Show that quadrilateral BRAT in the figure alongside is a:
3.2.1 parallelogram3.2.2 rectangle3.2.3 square
3.3Prove that RUCK is a rhombus.
3.4 Quadrilateral KILN is a kite. Show why this is so.
Show:
Not recommended
87
A(3;2)
B(-1;5)
C(1;-2)
D(x;y)
x
y
P(4;8)
S(-11;-3)O(-2;-1)
T(-14;4)x
y M(-1;8)
I(-4;-6)
A(8;-1)
L(-13;3)
y
x
C(2;7)
S(-12;-5)
A(4;-1)
T(-14;3)x
y
L(-4;-4)
C(6;6)
U(7;-5)
K(-5;7) y
x
F(-7;-4)
I(4;12)
A(-10;9)
L(6;-2)
x
yF(5;9)A(-8;8)
L(-2;-2)Y
x
y
-10
3.5 Conclude as accurately as possible what type of quadrilateral each of the following is,showing full details of how you came to your conclusions:
3.5.1 3.5.2
3.5.3 3.5.4
3.5.5 3.5.6
Finding the fourth vertex of a parallelogram.
**e.g.7 A(3;2), B(-1;5), C(1;-2) and D(x ; y) are the vertices ofparallelogram ABCD.(a) Find the coordinates of the midpoint of AC.(b) Complete: The diagonals of a parallelogram …(c) Give the numerical coordinates of the midpoint
of BD.(d) Hence find the values of x and y.
» Answers (a)
1 2 1 2;2 2
3 1 2 2;
2 2
2;0
x x y y
(b) … bisect each other.(c) midpoint BD midpoint AC 2;0
88
Q(-6;5)P(2;7)
R(4;-3)S(x;y)
x
y
Q(-6;5)P(2;7)
R(4;-3)
S(x;y)
x
y
y
x
A
B(3;4)
C(4;-1)
D(-3;-3)
(d)
1 5; 2;0
2 2
1 4 5
5 0 5
D 5; 5
x y
x x
y y
**e.g.8 Given points P(2;7), Q(-6;5) and R(4;-3), find the coordinates of S, the fourth vertex of parallelogram 8.1 PQRS 8.2 PSQR.
» Answers 8.1 PR and QS are the diagonals. midpoint PR midpoint QS
2 4 7 3 6 5; ;
2 2 2 2
6;4 6; 5
6 6 12
5 4 1
x y
x y
x x
y y
S(12; 1) 8.2 PQ and SR are the diagonals.
midpoint PQ midpoint SR
2 6 7 5 4 3; ;
2 2 2 2
4 32;6 ;
2 2
4 4 8
3 12 15
x y
x y
x x
y y
S( 8;15)
… a quicker method to find the fourth vertex of a parallelogram (and logical!)
This is called the vector method as it uses the same principle as vectors, a concept used in physics.
Consider this parallelogram:
Since DA//CB and DA CB, this means that D to A is the same as C to B.i.e. D A C B … 1 left and 5 up.1 left and 5 up from D 3; 3 is A 3 1; 3 5 A 4;2
It works the other way too:i.e. B A C D … 7 left and 2 down.7 left and 2 down from B 3;4 is A 3 7;4 2 A 4;2
89
A(-2;5)
B(5;7)
D
C(4;0) D(-2;6)
A
C(2;3)
D(8;9) B(-2;-8)
C
A(3;-14)
D(5;-10)
R
Q
P
S
Q
R
P
S
H
y
x
E
L
P
T
**e.g.9 Given points P(2;7), Q(-6;5) and R(4;-3), find the coordinates of S, the fourth vertex of parallelogram 9.1 PQRS 9.2 PQSR.
» Answers 9.1 P S Q R i.e. 10 right, 8 down
S 2 10;7 8 12; 1
9.2 R S P Q i.e. 8 left, 2 down
S 4 8; 3 2 4; 5
§ Exercise 4
4.1 M(-3;7), N(4;3), O(1;-5) and P(x ; y) are the vertices ofparallelogram MNOP.4.1.1 Find the coordinates of the midpoint of MO.4.1.2 Give the numerical coordinates of the midpoint of NP.4.1.3 Hence find the values of x and y.
4.2 Use the fact that the diagonals of a parallelogram bisect each other to find the fourth vertexof each of the following parallelograms ABCD:
4.2.1 4.2.2 4.2.3
4.3 Use the vector method to find the fourth vertex of each of the parallelograms in question 4.2.
4.4 Find the fourth vertex of parallelogram ABCD:
4.4.1 A 3;9 ; B 5; 1 ;C 8;4 ;D ;x y
4.4.2 A 5; 3 ; B ; ;C 2; 5 ; D 6;7x y
4.4.3 A ; ; B 3;7 ;C 2;0 ; D 8; 6x y
4.5 Show that P 3; 1 ; I 2;4 ;C 3;0 and K 2; 5 are the vertices of parallelogram PICK.
4.6 Prove that quadrilateral RIGH is a rectangle, given R 3; 1 ;I 0;8 ;G 6;6 and H 3; 3 .
4.7 H ; ; E 4; 3 ;L 4; 1 and P 6;3x y are the vertices of parallelogram HELP.
4.7.1 4.7.1.1 Determine the gradients of LE and PH.4.7.1.2 Calculate the value of x.
4.7.2 4.7.2.1 Use the distance formula to calculate the length of LE.
4.7.2.2 Hence determine the value of y.4.7.3 4.7.3.1 Determine the equation of PE.
4.7.3.2 Hence determine the value of t if T 5; t lies on PE.
90
SQUARE P P P P P P P P P P P P P P P PRHOMBUS P P P P P P X P P P P P X P P X
RECTANGLE X X P P P P P P P P X X P X X XPARALLELOGRAM X X P P P P X P P P X X X X X X
TRAPEZIUM X X X P X P X X X X X X X X X XKITE X P X X X P X P X X P X X P X X
QUADRILATERAL X X X X X P X X X X X X X X X X
All
side
s eq
ual
Tw
o pa
irs
of a
djac
ent
side
s eq
ual
Opp
osit
e sid
es e
qual
A p
air
of si
des
para
llel
Opp
osit
e si
des
para
llel
Inte
rior
ang
le s
um 3
60º
All
angl
es 9
0º
A p
air
of e
qual
opp
osit
e an
gles
Both
pai
rs o
f op
posi
te
angl
es e
qual
Dia
gona
ls b
isec
t ea
ch
othe
r
A d
iago
nal b
isec
ted
at 9
0º
Both
dia
gona
ls b
isec
ted
at
90º
Diag
onal
s eq
ual
Dia
gona
l bise
cts
a pa
ir o
f op
posi
te a
ngle
sD
iago
nal b
isec
ts b
oth
pair
s of
oppo
site
ang
les
All
angl
es b
etw
een
diag
onal
and
sid
es 4
5º
y x
A
VR
Y
4.8 Using points V 3; 4 ;A 1; 7 ; R 2; 5 and Y 0; 2 : 4.8.1 Calculate the lengths of VR and AY, leaving you answer in surd form.4.8.2 Determine the coordinates of M, the midpoint of AY.4.8.3 Prove that VMAY.4.8.4 Prove that V, M and R are co-linear.4.8.5 Show that M is the midpoint of VR.4.8.6 State, with reason, what type of quadrilateral VARY is.
QUADRILATERALS: Answers to exercises
Exercise 1
Exercise 2
2.1 50 2.2 60 2.3 45 ;90 ;90 2.4 42 ;116 ;42 ;58 2.5 25;75 2.6 60;30
2.7 7;7 2.8 5 2.9 15 ;105 2.10 148 ;16 2.11 96 2.12 56
Exercise 3
3.1 SA
TL SA
6 4 1
6 2 24 2 1
5 1 2SA//TL
m
m m
ST
AL ST
4 22
2 16 4
26 5
ST//AL
m
m m
SALT is a parallelogram
2 22
2 22 2
SA 2 6 4 6 20
TL 1 5 2 4 20 SA
SA TL
2 22
2 22 2
ST 2 1 4 2 5
AL 6 5 6 4 5 ST
ST AL
SALT is a parallelogram
2 5 4 4 7
Midpoint SL ; ;42 2 2
6 1 6 2 7Midpoint AT ; ;4 Midpoint SL
2 2 2
SALT is a parallelogram
3.2 3.2.1
7 7 2 0Midpoint TR ; 0;1
2 2
1 1 6 8Midpoint AB ; 0;1 Midpoint TR
2 2
BRAT is a parallelogram
3.2.2 TB BR
8 2 8 01
1 7 1 7TB BR
m m
(proving quad has a right angle) TB BR
8 2 8 01
1 7 1 7BRAT is a rectangle
m m
3.2.3 TR AB
2 0 8 61
7 7 1 1TR AB
m m
(Proving diagonals perpendicular) TR AB
2 0 8 61
7 7 1 1BRAT is a square
m m
(Proving diagonals bisect)
91
3.3
4 2 8 4Midpoint CR ; 1;2
2 2
9 7 2 6Midpoint UK ; 1;2 Midpoint CR
2 2
RUCK is a parallelogram
CR UK
8 4 2 61
4 2 9 7CR UK
m m
TR AB
2 0 8 61
7 7 1 1RUCK is a rhombus
m m
3.4
2 22
2 22 2
IK 12 10 9 3 148
IL 12 0 9 7 148 IK
IK IL
2 22
2 22 2
NK 10 1 3 4 122
NL 1 0 4 7 122 NK
NK NL
3.5 3.5.1 Definitely not a parallelogram! Test for trapezium:
TP SO TP
4 8 2 3 1 2TP//SO
14 4 9 2 11 9m m m
trapezium
3.5.2 Could be parallelogram. Test using midpoints of diagonals:13 8 3 1 5
Midpoint LA ; ;12 2 2
1 4 8 6 5Midpoint MI ; ;1 Midpoint LA parallelogram
2 2 2
Could be rhombus. Test using gradients of diagonals:
LA MI
3 1 8 61 CR UK
13 8 1 4m m
3.5.3 Could be rectangle. Test for parallelogram using midpoints of diagonals:
14 4 3 1Midpoint TA ; 5;1
2 2
2 12 8 6Midpoint CS ; 5;1 Midpoint TA parallelogram
2 2
Test for rectangle using gradients of CT, TS:
CT TS
7 3 3 51 CT TS
2 14 14 12m m
3.5.4 Could be rhombus. Test for parallelogram using midpoints of diagonals:
5 7 7 5Midpoint KU ; 1;1
2 2
6 4 6 4Midpoint CL ; 1;1 Midpoint KU parallelogram
2 2
Test for rhombus using gradients of CL, KU:
CL KU
6 4 7 51 CL KU
6 4 5 7m m
3.5.5 midpoint AL midpoint IF; no lines //; no adjacent sides =3.5.6 Could be kite. Check lengths of adjacent sides:
2 2 2 22 2
2 2 2 22 2
AY 8 10 8 0 68 LY 10 2 2 0 68 AY LY
AF 8 5 8 9 170 LF 5 2 9 2 170 AF LF
Exercise 4
4.1.1 1;1 4.1.2 1;1 4.1.3 6; 1
4.2.1 3; 2 4.2.2 4;12 4.2.3 0; 4
4.3.1 3; 2 4.3.2 4;12 4.3.3 0; 4
4.4.1 0; 6 4.4.2 1;9 4.4.3 7;1
4.7.1.1 undefined 4.7.1.2 6 4.7.2.1 2 4.7.2.2 1y 4.7.3.1 3 35 5y x 4.7.3.2 12
5
4.8.1 26 ; 26 4.8.2 912 2; 4.8.6 square
(Proving diagonals bisect)
(Proving adjacent sides equal)
KILN is a kite
rectangle
rhombus
quadrilateral
kite