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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
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
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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.