CONSTRUCTING A

QUADRILATERAL

WE SHALL LEARN HOW TO CONSTRUCT A UNIQUE QUADRILATERAL BY FOLLOWING METHODS :-

When four sides and one diagonal are given. When two diagonals and three sides are given. When two adjacent sides and three angles are

given. When three sides and two included angles are

given. When other special properties are known.

CONSTRUCTING A QUADRILATERAL WHEN FOUR SIDES AND ONE DIAGONAL ARE GIVEN

Example : Construct a quadrilateral PQRS where PQ = 4cm, QR = 6cm, RS= 5cm, PS = 5.5cm and PR = 7cm

7 cm

4 cm6 cm

5.5 cm 5 cm

S

R

Q

P

4 cm6 cm

RP

Step 1 :-Construct a triangle PQR using

SSS criterion.

7 cm

Q

STEP 2 :-WITH P AS CENTRE DRAW AN ARC OF RADIUS 5.5 CM .

4 cm6 cm

RP7 cm

Q

STEP 3 :-NOW, WITH R AS CENTRE DRAW AN ARC OF RADIUS 5 CM INTERSECTING ANOTHER ARC AT POINT S

4 cm6 cm

RP7 cm

Q

S

STEP 4 :-NOW JOIN PS AND RS.

PQRS is the required quadrilateral.

4 cm6 cm

S

P7 cm

Q

R

CONSTRUCTING A QUADRILATERAL WHEN TWO DIAGONALS AND THREE SIDES ARE GIVEN.

Example : Construct a quadrilateral ABCD, given that BC =4.5 cm, AD = 5.5 cm, CD = 5 cm, AC = 5.5 cm and diagonal BD = 7 cm.

C

D

A

B7 cm5 cm

5.5 cm

5.5c

m

4.5 cm

STEP 1 :-DRAW TRIANGLE ACD USING SSS CRITERION

C

D

A5.5 cm

5 cm5.5 cm

STEP 2 :-WITH D AS CENTRE DRAW AN ARC OF RADIUS 7 CM.

C

D

A5.5 cm

5 cm5.5 cm

STEP 3 :-NOW, WITH C AS CENTRE DRAW AN ARC OF RADIUS 4.5 CM INTERSECTING ANOTHER ARC AT POINT B.

C

D

A5.5 cm

5 cm5.5 cm

B

STEP 4 :-JOIN AB, DB AND CB

C

D

A5.5 cm

5 cm5.5 cm

ABCD is required triangleB

4 cm7 cm

CONSTRUCTING A QUADRILATERAL WHEN TWO ADJACENT SIDES AND THREE ANGLES ARE KNOWN

120

120 S and 105 I ,75 M cm, 6.5 IS cm, 3.5 MI whereMIST ralquadrilate

aConstruct : Example M

T

S

I105

75

6.5 cm

3.5

cm

.105 MIXan make andcm 3.5 MI line a Draw

-: 1 Step

105

3.5 cmM I

X

.120 ISYmake Now S.point that name

and 5.6 IX Measure-: 2 Step

3.5 cm

105

M I

X120

Y

S

6.5

cm

75 IMZ make Now-: 3 Step

3.5 cm

105

M I

X

120

Y

75

Z

S

STEP 4 :-MARK THAT POINT T WHERE SY AND MZ MEET.

MIST is the required Quadrilateral.

T

3.5 cm

105

M I

X

120

Y

75

Z

S

CONSTRUCTING A QUADRILATERAL WHEN THREE SIDES AND TWO INCLUDED DIAGONALS ARE GIVEN.

.80 C and 105 B cm, 6.5 CD

cm, 5 Bc cm, 4 AB whereABCD, ralQuadrilate

aConstruct -:

Example

A B

CD

4 cm

5 cm

6.5 cm

BX. along105 of an Draw andcm 5 BC line a Draw

-: 1 Step

5 cmB C

105

X

A.point that name and cm 4 BX Measure

-: 2 Step

5 cmB C

105

X

A

4 cm

80 BCY make Now

-: 3 Step

5 cmB C

105

XY

4 cm

80

D.point at CY cutsthat cm 6.5length of arcan Draw

centre, as C with Now,-: 4 Step

5 cmB C

105

X

4 cm

80

YD

6.5 cm

ADJoin -:Step5

ABCDis the required

Quadrilateral.

5 cmB C

105

X

4 cm

80

Y

D

6.5 cm

CONSTRUCTING A QUADRILATERAL WHEN OTHER SPECIAL PROPERTIES ARE KNOWN.

Some examples are :-1. Draw a square of side 5 cm.2. Construct a rhombus ABCD

where AC = 6 cm, and BD = 7 cm.

EXAMPLE 1 :-DRAW AN SQUARE ABCD OF SIDE 5 CM. Initially it appears

that only one measurements has been given. Actually we have many more details with us, because the figure is special quadrilateral namely a square. We know that each of its angles is a Right angle.

90

9090

4.5 cm

4.5 cm

4.5

cm

4.5 cm

90

A B

CD

EXAMPLE 2 :-CONSTRUCT A RHOMBUS ABCD WHERE AC = 6 CM, AND BD = 7 CM. As we know the diagonals of

Rhombus are Perpendicular bisectors of one another.

So, first we draw AC = 7 cm and construct its perpendicular bisector. Let them meet at O. cut off 3 cm lengths on either side of the drawn bisector . You get now B and D.

COA

B

D

7 cm

6 cm