Types of quadrilateral Name Shape Propertiesdanielmaths.weebly.com/uploads/5/1/2/1/5121151/grade_10...79 Types of quadrilateral Name Shape Properties 1. Quadrilateral 4 sides (quad.)

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:


4 4 6 2Midpt AC ; 0;2

2 2


2 2


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:


6 8 1 1Midpt AC ; 1;0

2 2


2 2


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


2 5 4 4 7

Midpoint SL ; ;42 2 2

6 1 6 2 7Midpoint AT ; ;4 Midpoint SL

2 2 2


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

Documents

Types of quadrilateral Name Shape Propertiesdanielmaths.weebly.com/uploads/5/1/2/1/5121151/grade_10...79 Types of quadrilateral Name Shape Properties 1. Quadrilateral 4 sides (quad.)