Construction

Construction of Quadrilaterals

The construction of square, rectangle, rhombus, parallelogram and trapezium are explained below:

1. Construction of a square when a side is given:

construction square

Here, ABCD is a square where AB = 8 cm.

Step 1: Draw AB = 8 cm. 

Step 2: Construct angle 90⁰ at both A and B. 

Step 3: Cut AX at 8 cm from A and name it D. Similarly, cut BY at 8 cm from B and name it C.

Step 4: Join C and D. 

Therefore, a square ABCD having a side 8 cm is formed.

2. Construction of a square when a diagonal is given:

Here, MNOP is a square whose diagonal NP = 5 cm.

Step 1: Draw NP = 5 cm.

Step 2: Draw 45⁰ angles at N and P.

Step 3: NX and PY intersect at M.

Step 4: Take an arc equal to MN or MP and draw two arcs downwards such that they intersect at C.

Step 5: Now, join N,O and P,O.

Therefore, a square MNOP having a diagonal NP = 5 cm is formed.

3. Construction of a rectangle when two adjacent sides are given:

Here, ABCD is a rectangle in which BC = 5 cm and AB = 4 cm.

Step 1: Draw BC = 5 cm.

Step 2: Construct 90⁰ angles at both points A and B.

Step 3: Take an arc of 4 cm and cut the BX at A and CY at D.

Step 4: Join A and B.

Therefore, a rectangle ABCD having BC = 5 cm and AB = 4 cm is formed.

4. Construction of a rectangle when diagonal and an angle made by them are given:

Here, ABCD is a rectangle in which AC = BD = 5 cm and an angle = 45⁰.

Step 1: Draw AC = 5 cm.

Step 2: Find the mid-point O of the line AC with the help of perpendicular bisector method.

Step 3: Construct 45⁰ angle at O. Produce the line XO straight upto Y on other side.

Step 4: Take an arc of 2.5 cm and cut OX at D and OY at B.

Step 5: Join A and B, B and C, C and D, D and A.

Therefore, a rectangle ABCD having AC = BD = 5 cm and an angle = 45⁰ is formed.

5. Construction of a Rhombus when a side and an angle between two adjacent sides are given:

Here, a rhombus MNOP with side MN = 5 cm and∠NMP = 60⁰.

Step 1: Draw MN = 5 cm.

Step 2: Draw ∠NMX = 60⁰ at M.

Step 3: Take measure of 5 cm and cut MX at P.

Step 4: Take a radius of 5 cm from N and P, draw two arcs such that they intersect at O.

Step 5: Join P and O, O and N.

Therefore, a rhombus MNOP is formed.

6. Construction of a Rhombus when the lengths of two diagonals are given:

Here, a rhombus ABCD having diagonal AC = 5 cm and BD = 6 cm.

Step 1: Draw the diagonal AC = 5 cm.

Step 2: Draw the perpendicular bisector XY of AC such that the mid-point is O.

Step 3: Take radius of 3 cm (half of diagonal BD) and from O, cut OY at D and OX at B.

Step 4: Joint A and B, B and C, C and D, D and A.

Therefore, a rhombus ABCD is formed.

7. Construction of a rhombus when a side and a diagonal are given:

Here, a rhombus ABCD where AB = 6 cm and AC = 8 cm.

Step 1: Draw AB = 6 cm.

Step 2: Keeping A center, take an arc of 8 cm.

Step 3: From B, take an arc of 6 cm and cut the previous arc and name it C.

Step 4: Join B and C, A and D.

Step 5: Again, taking A as centre, draw an arc of 5.5 cm upright and take another same arc from C and cut the previous arc and name it.

Step 6: Join C and D, D and A.

Therefore, a rhombus ABCD is formed.

8. Construction of a parallelogram when two adjacent sides and an angle contained by them are given:

Here, a parallelogram ABCD in which BC = 6 cm, CD = 5 cm and∠BCD = 120⁰.

Step 1: Draw BC = 6 cm.

Step 2: Construct an angle of 120⁰ at C, i.e.∠BCX = 120⁰.

Step 3: Take an arc of 4.5 cm and cut CX at D.

Step 4: With centre at B, draw an arc upright with radius of 4.5 cm.

Step 5: Likewise, with centre at D, draw another arc with radius of 5 cm.

Step 6: Join A and D, A and B.

Therefore, a parallelogram ABCD is formed.

9. Construction of a parallelogram when a base side, diagonal and angle contained by them are given:

Here, a parallelogram MNOP where base MN = 6 cm, diagonal MO = 8 cm and∠NMO = 30⁰.

Step 1: Draw MN = 6 cm.

Step 2: Construct an angle of 30⁰ at M (i.e. ∠NMX = 30⁰).

Step 3: Take a radius of 8 cm and cut MX at O from M.

Step 4: Join N and O.

Step 5: With centre at O, take radius of 6 cm and draw an arc.

Step 6: Likewise with centre at M, take a radius equal to NO and cut the previous arc. Name the intersecting points as P.

Step 7: Join O and P, P and M.

Therefore, a reqiured parallelogram MNOP is formed.

10. Construction of a parallelogram when a side and two diagonals are given:

Here, PQ = 6 cm and diagonals PR = 9 cm and QS = 6.5 cm.

Step 1: Draw a base line PQ = 6 cm.

Step 2: Take a radius of 5 cm and draw an arc up from point P.

Step 3: Similarly, take a radius of 5 cm and draw an arc up from Q. Such that it meets the previous arc. Name the point O.

Step 4: Join the point O to P and Q.

Step 5: Produce PO to R such that PO = OR. And produce QO to S such that QO =OS.

Step 6: Join P and S, S and R, R and Q.

Therefore, the required parallelogram PQRS is formed.

11. Construction of a parallelogram when two diagonals and an angle contained by them are given:

Here, PR = 6 cm, QS = 8 cm and angle between these two diagonals is 45⁰.

Step 1: Draw PR = 6 cm.

Step 2: Draw the bisector of PR and find the mid-point O.

Step 3: At O, draw an angle of 45⁰.

Step 4: Take a radius of 4 cm (half of QS) and cut OX at S and OY at Q.

Step 5: Join P and Q, Q and R, R and S, S and P.

Therefore, the required parallelogram PQRS is formed.

12. Construction of a trapezium when two adjacent sides and two angles are given:

Here, AB = 6 cm, BC = 5 cm, ∠DAB = 60⁰, ∠BCD = 90⁰ and AD||BC.

Step 1: Draw a base line AB = 6 cm.

Step 2: Construct an angle of ∠BAX =60⁰.

Step 3: Since, BC||AD, the angle at B should be 120⁰. So, construct an angle of 120⁰. (i.e. ∠ABY = 120⁰).

Step 4: Taking a radius of 5 cm, mark the point C on BY from the point B.

Step 5: Construct an angle of 90⁰at C. Thus formed line CD meets the pervious line AX at D.

Therefore, the required trapezium ABCD is formed.

13. Construction of a trapezium ABCD when two sides, a diagonal and an angle made by the diagonal with the given one side are given:

Here, AB = 6 cm, diagonal BD = 8 cm, ∠BAD = 60⁰.

Step 1: Draw a base line AB = 6 cm.

Step 2: Construct an angle of 60⁰ (i.e. ∠BAX) at A.

Step 3: Taking a radius of 8 cm, mark the point D on AX from Q.

Step 4: Join B and D.

Step 5: Since AB||CD, alternate angles are equal. So, construct an angle of 60⁰ at C with base AC to make an equal alternate∠XDC.

Step 6: Take a radius of 7 cm and from point D draw an arc to cut DY at C.

Step 7: Join C and B.

Therefore, the required trapezium ABCD is formed.

14. Construction of a trapezium PETU when three sides and an angle are given:

Here, PE = 8 cm, diagonal ET = 6 cm, TU = 4 cm and ∠PET = 60⁰ such that PE||TU.

Step 1: Draw a base line segment PE = 8 cm.

Step 2: Draw an angle∠PEX = 60⁰ at point E.

Step 3: Taking an arc of 6 cm from E, mark point T on EX.

Step 4: At point T, draw an∠ETY = 120⁰.

Step 5: Taking an arc of 4 cm from T, mark point U on TY.

Step 6: Join U and P.

Therefore, the required trapezium PETU is formed where PE||TU.