Constructions

Steps of construction:

(i) Draw a line segment AB=8.6 cm.

(ii) Draw a circle with centre A with radius 1/2 AB on upper and lower side of AB.

Similarly, Draw a circle with centre B with radius 1/2 AB on upper and lower side of AB which intersect above a circle at M and N respectively.

(iii) Draw line segment MN which intersect AB at point O.

On measuring,

AO=BO=1/2 AB=4.3

Steps of construction:

(i) Draw a line segment AB=5.8 cm.

(ii) Draw a circle with centre A with radius more than 1/2 AB.

Similarly, Draw a circle with centre B with same radius which intersect above circle at M and N respectively.

(iii) Draw line segment MN which intersect AB at point O.

On measuring,

m∠AOM=m∠BOM=90°

Steps of construction:

(i) Draw a circle with centre O and radius 5 cm.

(ii) Draw its chord AB.

(iii) Draw a circle with centre A with radius more than 1/2 AB.

Similarly, Draw a circle with centre B with same radius 1/2 AB which intersect above circle at M and N respectively.

(iv) Draw line segment MN which intersect AB at point P.

We can see that, perpendicular bisector MN of AB passes through centre O.

Steps of construction:

(i) Draw a circle with centre O.

(ii) Draw its two chords AB and CD such that AB is not parallel to CD.

(iii) Draw a circle with centre A with radius more than 1/2 AB.

Similarly, Draw a circle with centre B with same radius which intersect above circle at M and N respectively.

(iv) Draw line segment MN which intersect AB at point P.

(v) Similarly, Draw a circle with centre C with radius more than 1/2 CD.

Similarly, Draw a circle with centre D with same radius which intersect above circle at X and Y respectively.

(vi) Draw line segment XY which intersect CD at point Q.

We can see that two bisectors MN and XY intersects at centre O.

Steps of construction:

(i) Draw a line segment AB=10 cm.

(ii) Draw a circle with centre A with radius more than 1/2 AB.

Similarly, Draw a circle with centre B with same radius which intersect above circle at M and N respectively.

(iii) Draw line segment MN which intersect AB at point O.

On measuring,

AO=BO=1/2 AB=5 cm

(iv) Draw a circle with centre A with radius more than 1/2 AO.

Similarly, Draw a circle with centre O with same radius which intersect above circle at P and Q respectively.

(v) Draw line segment PQ which intersect AO at X.

On measuring,

AX=XO=1/2 AO=2.5 cm

Steps of construction:

(i) Draw a line segment AB.

(ii) Draw a circle with centre A with radius more than 1/2 AB.

Similarly, Draw a circle with centre B with same radius which intersect above circle at M and N respectively.

(iii) Draw line segment MN which intersect AB at point O.

On measuring,

AO=BO=1/2 AB

(iv) Draw a circle with centre A with radius more than 1/2 AO.

Similarly, Draw a circle with centre O with same radius which intersect above circle at P and Q respectively.

(v) Draw line segment PQ which intersect AO at X.

On measuring,

AX=XO=1/2 AO=1/4 AB

Steps of construction:

(i) Draw a line segment AB.

(ii) Draw a circle with centre A with radius more than 1/2 AB.

Similarly, Draw a circle with centre B with same radius which intersect above circle at M and N respectively.

(iii) Draw line segment MN which intersect AB at point O.

On measuring,

AO=BO=1/2 AB

(iv) Draw a circle with centre A with radius more than 1/2 AO.

Similarly, Draw a circle with centre O with same radius which intersect above circle at P and Q respectively.

(v) Draw line segment PQ which intersect AO at X.

On measuring,

AX=XO=1/2 AO=1/4 AB

∴ XB=XO+BO=1/4 AB+1/2 AB=3/4 AB

The steps of the required construction are:

1) Draw an arbitrary angle ∠BAC.

2) Taking A as the center and any radius draw an arc which intersects AB and AC at point D and E respectively.

3) Draw a line segment QR of arbitrary length. With Q as the center and Radius AD, draw a circular arc, intersecting QR at S.

4) Taking S as the center and radius DE, draw another circular arc, intersecting the previous arc at P. Join QP.

5) ∠RQP=∠BAC

The steps of the required construction are:

1) Draw an angle ∠BAC=150° using protractor.

2) Taking B as the center draw an arc of any radius greater than . Now, Taking C as the center and the keeping the same radius, draw another arc, intersecting the previous arc at D. Join AD.

3)

The steps of the required construction are:

1) Draw an angle ∠BAC=108° using a protractor.

2) Taking B as the center draw an arc of any radius greater than . Now, Taking C as the center and the keeping the same radius, draw another arc, intersecting the previous arc at D. Join AD.

3)

The steps of the required construction are:

1) Draw an angle ∠BAC=90° using a protractor.

2) Taking B as the center draw an arc of any radius greater than . Now, Taking C as the center and the keeping the same radius, draw another arc, intersecting the previous arc at D. Join AD.

3)

The steps of the required construction are:

1) Draw a line segment and choosing an arbitrary point A on it as the center and with any radius, draw a semi-circle, intersecting the line segment at point B and C. Choosing any arbitrary point D on the semi-circle, join AD. Thus, ∠BAD and ∠CAD are a linear pair of angles.

2) Taking B as the center draw an arc of any radius greater than . Now, Taking D as the center and the keeping the same radius, draw another arc, intersecting the previous arc at E. Join AE. Similarly, Taking C as the center draw an arc of any radius greater than . Now, Taking D as the center and the keeping the same radius, draw another arc, intersecting the previous arc at F. Join AF.

3) AE and AF are angle bisectors of ∠BAD and ∠CAD respectively. Measure ∠FAE using a protractor. It comes out to be 90°.

The steps of the required construction are:

1) Draw two intersecting line segments, intersecting at E. Taking E as center and any radius, draw a circle, intersecting the line segments at A, B, C and D. Thus ∠AEC and ∠BED are vertically opposite angles.

2) Taking A as the center draw an arc of any radius greater than . Now, Taking C as the center and the keeping the same radius, draw another arc, intersecting the previous arc at F. Join EF. Similarly, Taking B as the center draw an arc of any radius greater than . Now, Taking D as the center and the keeping the same radius, draw another arc, intersecting the previous arc at G. Join EG.

3) Measure ∠FEG using a protractor. It comes out to be 180°. Hence, FEG is a straight line.

The steps of the required construction are:

1) Draw a line segment AB. Keeping A as the center and any radius draw a semicircle, intersecting AB at point C.

2) Keeping C as the center and radius AC, draw an arc, cutting the semicircle at point D. Keeping D as the center and radius AC, draw an arc, cutting the semicircle at point E.

3) Taking D as the center draw an arc of any radius greater than . Now, Taking E as the center and the keeping the same radius, draw another arc, intersecting the previous arc at F. Join AF.

4) ∠BAF=90°.

The steps of the required construction are:

1) Draw a line segment BC. Taking any arbitrary point A on line segment BC as the center and any radius draw a semicircle, intersecting BC at points D and E.

2) Taking D as the center draw an arc of any radius greater than . Now, Taking E as the center and the keeping the same radius, draw another arc, intersecting the previous arc at F. Join AF, which intersects the semicircle at point G.

3) Taking G as the center draw an arc of any radius greater than . Now, Taking E as the center and keeping the same radius, draw another arc, intersecting the previous arc at H. Join AH.

4) ∠BAF=135°.

The steps of the required construction are:

1) Draw ∠BAC=72° using a protractor. Using A as the center and radius less than AB and AC, draw a circular arc, intersecting AB and AC at D and E respectively.

2) Taking D as the center draw an arc of any radius greater than . Now, Taking E as the center and the keeping the same radius, draw another arc, intersecting the previous arc at F. Join AF, which intersects the arc DE at point G. ∠BAF=36°.

3) Taking G as the center draw an arc of any radius greater than . Now, Taking E as the center and keeping the same radius, draw another arc, intersecting the previous arc at H. Join AH.

4) ∠BAH=54°.

i) The steps of the required construction are:

1) Draw a line segment BC. Taking any arbitrary point, A on line segment BC as the the center and any radius draw a semicircle, intersecting BC at points D and E.

2) Taking D as the center draw an arc of any radius greater than . Now, Taking E as the center and the keeping the same radius, draw another arc, intersecting the previous arc at F. Join AF, which intersects the semicircle at point G.

3) Taking D as the center draw an arc of any radius greater than . Now, Taking G as the center and the keeping the same radius, draw another arc, intersecting the previous arc at H. Join AH, which intersects the semicircle at point I.

4) ∠DAI=45°.

Justification:

Since DAE is a straight line therefore ∠DAE=180°.

Consider ∆EAF and ∆DAF

AE=AD (Radius of semi-circle)

EF=DF (By construction)

AF=AF (Common side)

Hence, By SSS criteria, ∆EAF≅∆DAF.

Therefore, by C.P.C.T. .

Consider ∆GAH and ∆DAH

AG=AD (Radius of semi-circle)

GH=DH (By construction)

AH=AH (Common side)

Hence, By SSS criteria, ∆GAH≅∆DAH.

Therefore, by C.P.C.T.

Hence, ∠DAI=∠DAH=45°.

ii) The steps of the required construction are:

The steps of the required construction are:

1) Draw a line segment BC. Taking any arbitrary point, A on line segment BC as the the center and any radius draw a semicircle, intersecting BC at points D and E.

2) Taking D as the center draw an arc of any radius greater than . Now, Taking E as the center and the keeping the same radius, draw another arc, intersecting the previous arc at F. Join AF, which intersects the semicircle at point G.

3) ∠DAG=90°.

Justification:

Since DAE is a straight line therefore ∠DAE=180°.

Consider ∆EAF and ∆DAF

AE=AD (Radius of semi-circle)

EF=DF (By construction)

AF=AF (Common side)

Hence, By SSS criteria, ∆EAF≅∆DAF.

Therefore, by C.P.C.T.

Hence, ∠DAG=∠DAF=90°.

i) The steps of the required construction are:

1) Draw a line segment AB. Keeping A as the center and any radius draw a semicircle, intersecting AB at point C.

2) Keeping C as the center and radius AC, draw an arc, cutting the semicircle at point D.

3) Taking C as the center draw an arc of any radius greater than . Now, Taking D as the center and the keeping the same radius, draw another arc, intersecting the previous arc at E. Join AE.

4) ∠BAE=30°.

ii) The steps of the required construction are:

1) Draw a line segment AB. Keeping A as the center and any radius draw a semicircle, intersecting AB at point C.

2) Keeping C as the center and radius AC, draw an arc, cutting the semicircle at point D. Keeping D as the center and radius AC, draw an arc, cutting the semicircle at point E.

3) Taking D as the center draw an arc of any radius greater than . Now, Taking E as the center and the keeping the same radius, draw another arc, intersecting the previous arc at F. Join AF, intersecting the semi-circle at G.

4) Taking D as the center draw an arc of any radius greater than . Now, Taking G as the center and the keeping the same radius, draw another arc, intersecting the previous arc at H. Join AH.

5) ∠BAH=75°.

iii) The steps of the required construction are:

1) Draw a line segment AB. Keeping A as the center and any radius draw a semicircle, intersecting AB at point C.

2) Keeping C as the center and radius AC, draw an arc, cutting the semicircle at point D. Keeping D as the center and radius AC, draw an arc, cutting the semicircle at point E.

3) Taking D as the center draw an arc of any radius greater than . Now, Taking E as the center and the keeping the same radius, draw another arc, intersecting the previous arc at F. Join AF, intersecting the semi-circle at G.

4) Taking E as the center draw an arc of any radius greater than . Now, Taking G as the center and the keeping the same radius, draw another arc, intersecting the previous arc at H. Join AH.

5) ∠BAH=105°.

iv) The steps of the required construction are:

1) Draw a line segment BC. Taking any arbitrary point A on line segment BC as the center and any radius draw a semicircle, intersecting BC at points D and E.

2) Taking D as the center draw an arc of any radius greater than . Now, Taking E as the center and the keeping the same radius, draw another arc, intersecting the previous arc at F. Join AF, which intersects the semicircle at point G.

3) Taking G as the center draw an arc of any radius greater than . Now, Taking E as the center and keeping the same radius, draw another arc, intersecting the previous arc at H. Join AH.

4) ∠BAF=135°.

v) The steps of the required construction are:

1) Draw a line segment AB. Keeping A as the center and any radius draw a semicircle, intersecting AB at point C.

2) Keeping C as the center and radius AC, draw an arc, cutting the semicircle at point D.

3) Taking C as the center draw an arc of any radius greater than . Now, Taking D as the center and the keeping the same radius, draw another arc, intersecting the previous arc at E. Join AE, intersecting the semi-circle at F.

4) Taking C as the center draw an arc of any radius greater than . Now, Taking F as the center and the keeping the same radius, draw another arc, intersecting the previous arc at G. Join AG.

5) ∠BAG=15°.

vi) The steps of the required construction are:

1) Draw a line segment BC. Taking any arbitrary point, A on line segment BC as the the center and any radius draw a semicircle, intersecting BC at points D and E.

2) Taking D as the center draw an arc of any radius greater than . Now, Taking E as the center and the keeping the same radius, draw another arc, intersecting the previous arc at F. Join AF, which intersects the semicircle at point G.

3) Taking D as the center draw an arc of any radius greater than . Now, Taking G as the center and the keeping the same radius, draw another arc, intersecting the previous arc at H. Join AH, which intersects the semicircle at point I.

4) Taking D as the center draw an arc of any radius greater than . Now, Taking I as the center and the keeping the same radius, draw another arc, intersecting the previous arc at J. Join AJ.

5) .

The steps of the required construction are:

1) Draw a line segment BC=3.6cm. Using a protractor, draw ∠CBD=60°. Join and extend BD.

2) Taking B as the center and radius=4.8cm, draw an arc, intersecting extended BD at point P. Join PC.

3) Taking P as the center and radius greater than , draw arcs on each side of PC. Now, taking C as the center and same radius, draw arcs, intersecting the previous arcs at points Q and R. Join and extend QR. Extended QR intersects line segment DB at point A. Join AC.

4) ∆ABC is the required triangle.

The steps of the required construction are:

1) Draw a line segment BC=4.5cm. Using a protractor, draw ∠CBD=45°. Join and extend BD.

2) Taking B as the center and radius=5.6cm, draw an arc, intersecting extended BD at point P. Join PC.

3) Taking P as the center and radius greater than , draw arcs on each side of PC. Now, taking C as the center and same radius, draw arcs, intersecting the previous arcs at points Q and R. Join and extend QR. Extended QR intersects line segment DB at point A. Join AC.

4) ∆ABC is the required triangle.

The steps of the required construction are:

1) Draw a line segment BC=3.4cm. Using a protractor, draw ∠CBD=45°. Join BD and extend it.

2) Taking B as the center and radius=1.5cm, draw an arc, intersecting BD at point P. Join PC.

3) Taking P as the center and radius greater than , draw arcs on each side of PC. Now, taking C as the center and same radius, draw arcs, intersecting the previous arcs at points Q and R. Join and extend QR. Extended QR intersects extended line segment DB at point A. Join AC.

4) ∆ABC is the required triangle.

The steps of the required construction are:

1) Draw a line segment BC=7cm. Using a protractor, draw ∠CBD=60°. Join and extend BD.

2) Taking B as the center and radius=12cm, draw an arc, intersecting extended BD at point P. Join PC.

3) Taking P as the center and radius greater than , draw arcs on each side of PC. Now, taking C as the center and same radius, draw arcs, intersecting the previous arcs at points Q and R. Join and extend QR. Extended QR intersects line segment DB at point A. Join AC.

4) ∆ABC is the required triangle.

The steps of the required construction are:

1) Draw a line segment DE=6.4cm. Using a protractor, draw ∠EDF=60° and ∠DEG=45°. Join DF and EG. Taking D as the center and any radius, draw an arc, intersecting DE at H and DF at I. Similarly, Taking E as the center and any radius, draw an arc, intersecting DE at J and EG at K.

2) Taking H as the center draw an arc of any radius greater than . Now, Taking I as the center and the keeping the same radius, draw another arc, intersecting the previous arc at L. Join and extend DL. Similarly, Taking J as the center draw an arc of any radius greater than . Now, Taking K as the center and the keeping the same radius, draw another arc, intersecting the previous arc at M. Join and extend EM, intersecting extended DL at A.

3) Taking D as the center and radius greater than , draw arcs on each side of AD. Now, taking A as the center and same radius, draw arcs, intersecting the previous arcs at points N and O. Join and extend NO. Extended NO intersects line segment DE at point C. Join AC. Similarly, Taking E as the center and radius greater than , draw arcs on each side of AE. Now, taking A as the center and same radius, draw arcs, intersecting the previous arcs at points P and Q. Join and extend PQ. Extended NO intersects line segment DE at point B. Join AB.

4) ∆ABC is the required triangle.

The steps of the required construction are:

1) Draw a line segment DE=12cm. Using a protractor, draw ∠EDF=60° and ∠DEG=45°. Join DF and EG. Taking D as the center and any radius, draw an arc, intersecting DE at H and DF at I. Similarly, Taking E as the center and any radius, draw an arc, intersecting DE at J and EG at K.

2) Taking H as the center draw an arc of any radius greater than . Now, Taking I as the center and the keeping the same radius, draw another arc, intersecting the previous arc at L. Join and extend DL. Similarly, Taking J as the center draw an arc of any radius greater than . Now, Taking K as the center and the keeping the same radius, draw another arc, intersecting the previous arc at M. Join and extend EM, intersecting extended DL at A.

3) Taking D as the center and radius greater than , draw arcs on each side of AD. Now, taking A as the center and same radius, draw arcs, intersecting the previous arcs at points N and O. Join and extend NO. Extended NO intersects line segment DE at point C. Join AC. Similarly, Taking E as the center and radius greater than , draw arcs on each side of AE. Now, taking A as the center and same radius, draw arcs, intersecting the previous arcs at points P and Q. Join and extend PQ. Extended NO intersects line segment DE at point B. Join AB.

4) ∆ABC is the required triangle.