Construction of Quadrilaterals
You already know that a quadrilateral is a 4-sided polygon. It is plane (2D) figure with 4 sides, 4 corners and 2 diagonals. A quadrilateral can be constructed if its any 5 measurements are given. Let us see what are these 5 measurements when we can construct a quadrilateral.
We can construct a quadrilateral:
(i) When its Four Sides and One Angle are Given
(ii) When its Three Sides and Two Included Angles are Given
(iii) When Four Sides and One Diagonal are Given
(iv) When Three Sides and Two Diagonals are Given
(v) When its Three Sides, One Diagonal and One Angle are Given
Construction of a Quadrilateral
When its Four
Sides and One Angle are Given
Example: Construct a
quadrilateral ABCD in which AB = 5 cm, BC = 4.5 cm, CD = 5 cm, DA = 6 cm and ∠ABC = 60°.
Steps of
Construction
1. Draw a rough sketch of the
quadrilateral ABCD.
2. Construct a line segment AB measuring
5 cm.
3. At A, construct an angle
measuring 60° and draw ray m.
4. Taking A as center, draw an
arc of radius 6 cm cutting m at D.
5. Taking B as center, draw an
arc of radius 4.5 cm and taking D as center draw an arc of radius 5 cm. Let
both the arcs cut at a point C.
6. Join CD and BC.
Thus, ABCD is the required
quadrilateral.
When its Three
Sides and Two Included Angles are Given
Example: Construct a
quadrilateral PQRS in which PQ = 3.2 cm, QR = 2.5 cm, PS = 3 cm, ∠SPQ = 45° and ∠PQR = 120°.
Steps of
Construction
1. Draw a rough sketch of
quadrilateral PQRS.
2. Construct a line segment PQ measuring
3.2 cm.
3. Taking P as center, construct
an angle 45° and draw a ray m.
4. Taking Q as center, construct
an angle measuring 120° and draw a ray n.
5. Cut off PS = 3 cm on ray m and QR = 2.5 cm
on ray n. Join RS.
Thus, PQRS is the required quadrilateral.
When Four Sides and One
Diagonal are Given
Example: Construct a
quadrilateral MATE in which MA = 5.2 cm, AT = 4.4 cm, TE = 3.2 cm, ME = 2.8 cm
and MT = 5.5 cm.
Steps of
Construction
1. Draw a rough sketch of the
quadrilateral MATE.
2. Construct a ray l and mark MA =
5.2 cm on it.
3. Taking M as center, draw two
arcs whose radii are 2.8 cm and 5.5 cm respectively.
4. Taking A as center, draw an
arc whose radius is 4.4 cm and let it cut the previous arc drawn from M which
measures 5.5 cm. Mark the point as T and join MT and AT.
5. From T, draw an arc of radius
3.2 cm and let it cut the arc drawn from M at E whose radius is 2.8 cm. Join ME
and ET.
Thus, MATE the required quadrilateral.
When Three Sides and Two Diagonals
are Given
Example: Construct a quadrilateral
LION in which LI = 3 cm, IO = 2.7 cm, ON = 4.1 cm, LO = 5.2 cm and NI = 5.4 cm.
Steps of
Construction
1. Draw a rough sketch of the
quadrilateral LION.
2. Construct a line segment LI measuring
3 cm.
3. Taking L as center, draw an
arc of radius 5.2 cm and by taking I as center, draw another arc of radius 2.7
cm cutting the earlier arc at O. Join LO and IO.
4. Taking O as center, draw an
arc of radius 4.1 cm and taking I as center, draw another arc of radius 5.4 cm
cutting the earlier arc at N.
5. Join ON, NI and LN.
Thus, LION is the required
quadrilateral.
When its Three Sides, One Diagonal and One Angle are Given
Example: Construct a
quadrilateral PQRS in which PQ = 4.5 cm, QR = 3.3 cm, PS = 2.9 cm, QS = 4.8 cm
and ∠PQR = 60°.
Steps of
Construction
1. Draw a rough sketch of quadrilateral
PQRS.
2. Construct a line segment PQ measuring
4.5 cm.
3. At Q, draw a ray m at 60° and cut
off QR = 3.3 cm from it.
4. Taking Q as center, draw an
arc of radius 4.8 cm and by taking P as center, draw an arc of radius 2.9 cm
cutting the previous arc at S.
5. Join PS, QS and SR.
Thus, PQRS is the required quadrilateral.
Construction of a Rectangle
When the Length of Adjacent Sides are Given
Example: Construct a
rectangle ABCD in which AB = 4.2 cm and BC = 3.4 cm.
Steps of
Construction
1. Draw a rough sketch of the
rectangle ABCD.
2. Construct a line segment AB measuring
4.2 cm.
3. At A and B, draw rays l and m respectively perpendicular
to AB.
4. Cut off AD = BC = 3.4 cm on
ray l
and
m.
5. Join BC, CD and AD.
Thus, ABCD is the required rectangle.
When the Length of a Side and a Diagonal are Given
Example: Construct a
rectangle EFGH in which EF = 4 cm and FH = 5 cm.
Steps of
Construction
1. Draw a rough sketch of the
rectangle EFGH.
2. Construct a line segment EF measuring
4 cm and draw perpendiculars EX and FY at E and F respectively.
3. Taking F as center, draw an
arc of radius 5 cm to cut off EX at H.
4. Taking E as center, draw
another arc of radius 5 cm to cut off FY at G.
5. Join FH, EG and HG.
Thus, EFGH is the required rectangle.
Construction of a Square
When the Length of its Edge is Given
Example: Construct a
square ABCD in which AB = 4.2 cm.
Steps of
Construction
1. Draw a rough sketch of the
square ABCD.
2. Construct a line segment AB measuring
4.2 cm.
3. At A and B, draw
perpendiculars AX and BY respectively.
4. Cut off BC = AD = 4.2 cm.
5. Join BC, CD and AD.
Thus, ABCD is the required square.
Whose One Diagonal is Given
Example: Construct a
square LATE in which LT = AE = 4 cm.
Steps of
Construction
1. Draw a rough sketch of the
square LATE.
2. Construct a line segment LT measuring
4 cm and draw a perpendicular bisector XY of LT which cuts at O.
3. With O as center and OL as
radius, draw a circle. Let the circle cut XY at A and E respectively.
4. Join LA, AT, TE and LE.
Thus, LATE is the required square.
Construction of a Parallelogram
Whose Two Adjacent Sides and the Included Angle are Given
Example: Construct a
parallelogram PQRS in which PQ = 4.7 cm, QR = 3.5 cm and ∠PQR = 45°.
Steps of
Construction
1. Draw a rough sketch of the
parallelogram PQRS.
2. Construct a line segment PQ measuring
4.7 cm.
3. At Q, construct a line QX at
an angle of 45°.
4. Cut off QR = 3.5 cm from QX.
5. Taking P as center, draw an
arc of radius 3.5 cm and taking R as center, draw an arc of radius 4.7 cm intersecting
the previous arc at S.
6. Join PS and RS.
Thus, PQRS is the required quadrilateral.
When
Two Adjacent Sides and One Diagonal are Given
Example:
Construct a parallelogram ABCD in which AB = 5 cm, BC = 4.5 cm and BD = 5.3 cm.
Steps of
Construction
1. Draw a rough sketch of the
parallelogram ABCD.
2. Construct a line segment AB measuring
5 cm.
3. Taking B as center, draw an
arc of radius 5.3 cm, and taking A as center, draw another arc measuring 4.5 cm
such that it cuts the previous arc at D. Join AD and BD.
4. Taking D as center, draw an
arc of radius 5 cm and taking B as center, draw another arc of radius 4.5 cm
cutting the previous arc at C.
5. Join BC and CD.
Thus, ABCD is the required parallelogram.
When One Side and Two Diagonals are Given
Example: Construct a
parallelogram LION in which LI = 3.5 cm, LO = 5 cm and NI = 3.8 cm.
Steps of
Construction
1. Draw a rough sketch of the
parallelogram LION.
2. Construct ∆LAI where LI =
3.5 cm, LA = 2.5 cm and IA = 1.9 cm since diagonals of a parallelogram bisect
each other.
3. Produce LA to O such that LA
= AO and produce IA to N such that IA = AN.
4. Join LA, NO, OI.
Thus, LION is the required parallelogram.
When Two Diagonals and the Angle between Them are Given
Example: Construct a
parallelogram ABCD in which AC = 4.8 cm, BD = 3.8 cm and ∠AOD = 60°.
Steps of
Construction
1. Draw a rough sketch of the
parallelogram ABCD.
2. Construct a line segment AO measuring
2.4 cm and produce it to C such that OC = OA, as diagonals of a parallelogram
bisect each other.
3. At O, construct ∠AOX = 60°.
4. Cut off OD = 1.9 cm from OX
and produce DO to B such that DO = OB.
5. Join AB, BC, CD and AD.
Thus, ABCD is the required parallelogram.
Construction
of a Rhombus
When its Two Diagonals are Given
Example: Construct a
rhombus PQRS in which PR = 4 cm and QS = 6 cm.
Steps of
Construction
1. Draw a rough sketch of the
rhombus PQRS.
2. Construct a line segment QS measuring
6 cm and draw its perpendicular bisector XY cutting it at O.
3. From O, draw PO = OR = 2 cm.
4. Join PQ, QR, RS and PS.
Thus, PQRS is the required rhombus.