Mathematics

# THEOREMS AND PROBLEMS ON PARALLELOGRAMS – EXERCISE 4.3.5 – Class IX

1. Construct parallelograms ABCD with the following measurements:

(a) AD = 4.2 cm, DC = 5.8 cm, ∟D = 43˚

(b) AB = 43. Cm, BC = 3.2 CM, ∟C= 120˚

(c) AD =4.2 cm, DC = 5.8 cm, AC = 3.4cm

(d) AD = 4.3 cm, DC = 5.7 cm, DB = 7cm

(e) AD = 6.8 cm , DB = 7.6 cm, AD = 6.4cm

Solution:

(a) AD = 4.2cm, DC = 5.8cm, ∟D = 43˚

Steps of construction:

1. Draw a line segment AD = 4.2cm.
2. Draw DC = 5.8cm such that ∠ADC = 43°
3. With A as center and 5.8cm as radius draw another arc to intersect the above at B.
4. Join AB and CB. ABCD is the required parallelogram

(b) AB = 43. Cm, BC = 3.2 CM, ∟C= 120˚

Steps of construction:

C = 120  A =120 (opposite angles of a parallelogram)

BC = 3.2cm  A = 3.2cm (opposite angles of a parallelogram)

1. Draw a line segment AB = 4.3 cm.
2. At A make an angle of 120° and draw AE.
3. In A cut off AD = 3.2cm on AE.
4. With B as center and 3.2cm as radius draw an arc.
5. With D as center and 4.3 cm as radius draw another arc to intersect the above at C.
6. Join BC and DC.

ABCD is the required parallelogram.

(c) AD =4.2 cm, DC = 5.8 cm, AC = 3.4 cm

Step of construction:

2. With A as center and 3.4 cm as radius draw an arc.
3. With D as center and 5.8cm as radius draw another arc to intersect the above arc
4. Join AC and DC.
5. With A as center and 5.8 cm radius draw an arc.
6. With C as center and 4.2cm as radius draw another arc to intersect the above at B.
7. Join AB and CB. ABCD is the required parallelogram.

(d) AD = 4.3 cm, DC = 5.7 cm, DB = 7cm

AD = 4.3 cm DC = 5.7 cm = AB

Step of construction:

2. With A as center and 4.3 cm as radius draw an arc.
3. With B as center and 7cm as radius, draw another arc to intersect the above at D.
5. With B as center and 4.3 cm as radius draw an arc.
6. With D as center and 5.7 cm as radius draw another arc to intersect the above at C.
7. Join DC and BC. ABCD is the required parallelogram

Diagonals AC and BD bisect each other at E

AE = EC = 6.8/2 = 34

AE = EC = 7.6/2 = 3.8

Step of construction:

2. With A as center and 3.4 cm as radius draw an arc.
3. With D as center and 3.8cm as radius draw another arc to intersect the above arc at E.
4. Join AE and produce to C such that EC = 3.4cm.
5. Join DE and produces to B such that EB = 3.8cm.
6. Join AB, BC and DC.

ABCD is the required parallelogram

1. Cons trust rectangle ABCD with the following data:

(a) AB = 4cm, BC = 6cm

Step of construction:

1. Draw AB = 4cm.
2. Construct a perpendicular to AB at B and cut off BC = 6cm.
3. With A as center and 6cm as radius draw an arc.
4. With C as center and 4cm as radius draw another arc to intersect the above arc at D.

ABCD is the required parallelogram.

(b) AB = 6cm , AC = 7.2 cm

Step of construction:

1. Draw AB = 6cm.
2. At B construct ABE = 90.
3. With A as center and 7.2 cm as radius draw an arc to intersect the above at C.
4. Join AC.
5. Construct BAF = 900.
6. With B as center and 7.2 cm as radius draw an arc to intersect the above at D.
7. Join CD.

ABCD is the required parallelogram.

1. Construct square ABCD.
2. a) Which has side – length 2 cm
3. b) Which has diagonal 6cm

Solution:

1. a) Which has side – length 2 cm

Step of construction:

1. Draw AB = 2cm
2. At A draw AE  AB
3. With A as center and 2cm as radius draw an arc to intersect AE at D.
4. With D as center and 2cm as radius draw an arc.
5. With B as center and 2cm as radius draw another arc to intersect the above at C.
6. Join BC and DC

ABCD is the required parallelogram.

1. b) Which has diagonal 6cm

Step of construction:

1. Draw AC = 6cm.
2. Draw the perpendicular bisecular of AC to intersect it at E.
3. With E as center and 3 cm as radius draw an arc to intersect the above at B and D.
4. Join AB, BC, CD, and DA.

ABCD is the required parallelogram.

1. Construct rhombus ABCD such that:
2. a) AB = 3.2 cm, AC =4.8 cm
3. b) AB = 4.4 cm, BD = 5.4 cm
4. c) AC = 7 cm, BD = 4 cm
5. d) BD =6.8cm, AC = 5.4 cm.

Solution:

1. a) AB = 3.2 cm, AC =4.8 cm

Step of construction:

1. Draw AC = 3.2 cm.
2. With A as center and 4.8 cm as radius draw an arc.
3. With B as center and 3.2 cm as radius draw another arc to intersect to above at C.
4. Join AC and BC.
5. With A as center and 3.2 cm as radius draw an arc.
6. With C as center and 3.2 cm as radius draw another arc to intersect the above at D.

ABCD is the required parallelogram.

1. b) AB = 4.4 cm, BD = 5.4 cm.

Step of construction:

1. Draw a AB = 4.4 cm.
2. With A as center and 4.4 cm as radius draw an arc.
3. With B as center and 5.4 cm as radius draw another arc to intersect to above at D.
5. With B as center and 4.4 cm radius draw an arc.
6. With D as center and 4.4 cm as radius draw another arc to intersect the above at C.
7. Join DC and BC.

ABCD is the required parallelogram.

1. c) AC = 7cm, BD = 4 cm

7cm

4cm E

Step of construction:

1. Draw AC = 7 cm.
2. Draw perpendicular bisector of AC to intersect or it at E.
3. Cut off ED = EB = 2 cm.
4. Join AB, BC, CD and DA.

1. d) BD = 6.8 cm, AC = 5.4 cm

Step of construction:

1. Draw BD = 6.8 cm.
2. Draw PQ the perpendicular bisector of BD to intersect or it at E.
3. Cut off EA = EC = 2.7 cm.
4. Join AB, BC, CD and DA.

ABCD is the required square.