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.
(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

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.

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.

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.