## Note on Construction

• Note
• Things to remember
• Exercise

Construction in geometry means to draw shapes, angles or lines accurately. Drawing of geometric items such as lines and circles using only compasses and straight edge.

These constructions use only compass, straightedge (i.e. ruler) and a pencil.

### Construction of Rectangle

When two adjacent sides are given.

Example:

Construct a rectangle ABCD in which AB = 6cm and BC = 4.5cm.

Steps of Construction:

1. Draw AB = 6cm
2. At B, construct∠PBA =90° From BP cut BC = 4.5cm
3. Taking C as the centre, draw an arc of radius 6cm and taking A as the centre, draw another arc of the radius 4.5cm to cut the previous arc at D.
4. Join AD and CD.

Thus, ABCD is the required rectangle.

### Construction of Square

When one side is given

Example:

Construct the square in which AB=5.5cm

Steps of construction;

1. Draw AB = 5.5cm.
2. At A, Construct∠PAB = 90°
3. From AP, cut AD = 5.5cm
4. Taking D as a centre, draw an arc of radius 5.5cm and taking B as centre draw another arc of radius 5.5cm to cut the previous arc at C.
5. Join BC and DC.

Thus, ABCD is the required square.

### Construction of a Parallelogram

When two adjacent sides and the included angle are given.

Example:

Construct a parallelogram ABCD in which BC = 5.3 cm, CD = 4.6 cm and∠C = 60°.

Steps of Construction:

1. Draw BC = 5.3cm
2. At C, construct∠PCB = 60° and from CP cut CD = 4.6cm.
3. Taking D as a centre, draw an arc of radius 5.3cm and taking B as centre draw one more arc of radius 4.6cm to cut the previous arc at the point.
4. Join AB and AD.

Thus, ABCD is the required parallelogram.

### Construction of a Rhombus (an equilateral parallelogram)

When one side and one angle are given.

Example:

Construct a rhombus in which AB = 5.5 cm and∠A =45°

Steps of construction:

1. Draw ab = 5.5cm
2. At A, construct∠PAB = 45°
3. From AP, cut AD =5.5cm
4. Taking D as a centre, draw an arc of radius 5.5 cm and taking B as a centre draw another arc of radius 5.5cm to cut the previous arc at the point C.
5. Join BC and DC.

### Construction of Regular Pentagon

In a regular pentagon, all sides are equal in length and each interior angle is 108°.In the figure, ABCDE is a regular pentagon.

Example:

Construct a regular pentagon ABCDE in which AB = 3cm.

Steps of construction:

1. Draw a line MN and cut off AB = 3cm on MN
2. At A and B draw∠WAB =∠XBA = 108° using a protractor.
3. Taking A as the centre, draw an arc radius 3cm to cut AW at E and taking B as the centre, draw an arc of radius 3cm to cut BX at C.
4. At C and E draw ∠BCZ =∠AEY = 108°
5. CZ and EY intersect at D.

Thus, ABCDE is the required regular pentagon.

### Construction of a Regular Hexagon

In the regular hexagon, all sides are equal in length and each interior angle is 120°.

Also, exterior angle =$$\frac{360°}{n}$$

= $$\frac{360°}{6}$$ = 60°

Example:

Construct a regular hexagon ABCDEF in which AB=4cm

Steps of Construction:

1. Draw a line PQ and cut off AB = 4cm on PQ.
2. At B, make an angle of QBR = 60°
3. From BR cut off BC = 4cm
4. At C, make an angle RCS = 60°
5. Cut off CD = 4cm from CS
6. At D, make an angle SDT = 60°
7. Cut off DE = 4cm from DT.
8. At E, make an angle TEU = 60°
9. Cut off EF = 4cm from EU
10. Join FA

Thus, ABCDEF is required regular hexagon.

### Construction of a Regular Octagon

In a regular octagon, all sides are equal in length and each interior angle is 135°

Example:

Construct a regular octagon having a side 5 cm

Steps of construction:

1. Draw a line PQ and cut off AB =- 5cm on PQ
2. At A and B draw∠MAB =∠ABN = 135°
3. Taking A and B as centres, draw arcs of radius 5 cm to cut AM and AN at H and C respectively.
4. At H and C, draw∠ AHQ =∠BCR = 135°
5. Taking H and C as centres, draw arcs of radius 5cm to cut HQ and CR at G and D respectively.
6. At G and D, draw∠HGX =∠CDY = 135°
7. Taking G and D as centres, draw arcs of radius 5cm to cut GX and DY at F and E respectively.
8. Join FE

Thus, ABCDEFGH is a regular octagon.

• Construction means,the drawing of geometric items such as lines and circles using only compasses and straightedge or ruler.
• Very importantly, you are not allowed to measure angles with a protector, or measure lengths with a ruler.
.

### Very Short Questions

Solution:

Steps of construction:-

(i) Draw AB = 4.8cm

(ii) At B, construct $$\angle$$PBA=90o

(iii) Taking A as centre, draw an arc of radius 6.2cm which cuts BP at the point C

(iv) Taking C as centre, draw an arc of radius 4.8 cm and taking A as centre draw another arc of radius equal to BC to cut the previous arc at the point D

(v) Join AD and CD

Thus, ABCD is the required rectangle.

Solution:

Steps of construction:

(i) Draw a straight line PR=7 cm

(ii) Find the midpoint O of PR.

(iii) Using a compass, draw a line XY at 45o.

(iv) With the length of OP cut arc

(v) Using scale join PQRS.

(vi) The required figure PQRS is completed.

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##### Suraj

How to make regular octagon inside circle