Practical Geometry-Solutions 14.5

CBSE Class –VI Mathematics
NCERT Solutions
Chaper 14 Practical Geometry (Ex. 14.5)

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Question 1. Draw AB¯¯¯¯¯¯¯$\overline{\text{AB}}$ of length 7.3 cm and find its axis of symmetry.

Answer: Axis of symmetry of line segment AB¯¯¯¯¯¯¯$\overline{\text{AB}}$ will be the perpendicular bisector of AB¯¯¯¯¯¯¯.$\overline{\text{AB}}.$ So, draw the perpendicular bisector of AB.

Steps of construction:

(i) Draw a line segment AB¯¯¯¯¯¯¯$\overline{\text{AB}}$ = 7.3 cm

(ii) Taking A and B as centres and radius more than half of AB, draw two arcs which intersect each other at C and D.

(iii) Join CD. Then CD is the axis of symmetry of the line segment AB.

Question 2. Draw a line segment of length 9.5 cm and construct its perpendicular bisector.

Answer: Steps of construction:

(i) Draw a line segment AB¯¯¯¯¯¯¯$\overline{\text{AB}}$ = 9.5 cm

(ii) Taking A and B as centres and radius more than half of AB, draw two arcs which intersect each other at C and D.

(iii) Join CD. Then CD is the perpendicular bisector of AB¯¯¯¯¯¯¯$\overline{\text{AB}}$.

Question 3. Draw the perpendicular bisector of XY¯¯¯¯¯¯¯¯$\overline{\text{XY}}$ whose length is 10.3 cm.

(a) Take any point P on the bisector drawn. Examine whether PX = PY.

(b) If M is the mid-point of XY¯¯¯¯¯¯¯¯$\overline{\text{XY}}$, what can you say about the lengths MX and XY?

Answer: Steps of construction:

(i) Draw a line segment XY¯¯¯¯¯¯¯¯$\overline{\text{XY}}$ = 10.3 cm

(ii) Taking X and Y as centres and radius more than half of XY, draw two arcs which intersect each other at C and D.

(iii) Join CD. Then CD is the required perpendicular bisector of XY¯¯¯¯¯¯¯¯$\overline{\text{XY}}$.

Now:

(a) Take any point P on the bisector drawn. With the help of divider we can check that PX¯¯¯¯¯¯¯=PY¯¯¯¯¯¯¯$\overline{\text{PX}}=\overline{\text{PY}}$.

(b) If M is the mid-point of XY¯¯¯¯¯¯¯¯,$\overline{\text{XY}},$ then MX¯¯¯¯¯¯¯¯=12XY¯¯¯¯¯¯¯¯$\overline{\text{MX}}=\frac{1}{2}\overline{\text{XY}}$.

Question 4. Draw a line segment of length 12.8 cm. Using compasses, divide it into four equal parts. Verify by actual measurement.

Answer: Steps of construction:

(i) Draw a line segment AB = 12.8 cm

(ii) Draw the perpendicular bisector of AB¯¯¯¯¯¯¯$\overline{\text{AB}}$ which cuts it at C. Thus, C is the mid-point of AB¯¯¯¯¯¯¯.$\overline{\text{AB}}.$

(iii) Draw the perpendicular bisector of AC¯¯¯¯¯¯¯$\overline{\text{AC}}$ which cuts it at D. Thus D is the mid-point of AC¯¯¯¯¯¯¯$\overline{\text{AC}}$ .

(iv) Again, draw the perpendicular bisector of CB¯¯¯¯¯¯¯$\overline{\text{CB}}$ which cuts it at E. Thus, E is the mid-point of CB¯¯¯¯¯¯¯$\overline{\text{CB}}$.

(v) Now, point C, D and E divide the line segment AB¯¯¯¯¯¯¯$\overline{\text{AB}}$ in the four equal parts.

(vi) By actual measurement, we find that

AD¯¯¯¯¯¯¯¯=DC¯¯¯¯¯¯¯=CE¯¯¯¯¯¯¯=EB¯¯¯¯¯¯¯=3.2$\overline{\text{AD}}=\overline{\text{DC}}=\overline{\text{CE}}=\overline{\text{EB}}=3.2$ cm

Question 5.With PQ¯¯¯¯¯¯¯$\overline{\text{PQ}}$ of length 6.1 cm as diameter, draw a circle.

Answer: Steps of construction:

(i) Draw a line segment PQ¯¯¯¯¯¯¯$\overline{\text{PQ}}$ = 6.1 cm.

(ii) Draw the perpendicular bisector of PQ which cuts, it at O. Thus O is the mid-point of PQ¯¯¯¯¯¯¯$\overline{\text{PQ}}$.

Taking O as centre and OP or OQ as radius draw a circle where diameter is the line segment PQ¯¯¯¯¯¯¯$\overline{\text{PQ}}$.

Question 6.Draw a circle with centre C and radius 3.4 cm. Draw any chord AB¯¯¯¯¯¯¯.$\overline{\text{AB}}.$ Construct the perpendicular bisector AB¯¯¯¯¯¯¯$\overline{\text{AB}}$ and examine if it passes through C.

Answer: Steps of construction:

(i) Draw a circle with centre C and radius 3.4 cm.

(ii) Draw any chord AB¯¯¯¯¯¯¯.$\overline{\text{AB}}.$

(iii) Taking A and B as centers and radius more than half of AB¯¯¯¯¯¯¯,$\overline{\text{AB}},$ draw two arcs which cut each other at P and Q.

(iv) Join PQ. Then PQ is the perpendicular bisector of AB¯¯¯¯¯¯¯.$\overline{\text{AB}}.$

This perpendicular bisector of AB¯¯¯¯¯¯¯$\overline{\text{AB}}$ passes through the centre C of the circle.

Question 7.Repeat Question 6, if AB¯¯¯¯¯¯¯$\overline{\text{AB}}$ happens to be a diameter.

Answer: Steps of construction:

(i) Draw a circle with centre C and radius 3.4 cm.

(ii) Draw its diameter AB¯¯¯¯¯¯¯$\overline{\text{AB}}$.

(iii) Taking A and B as centers and radius more than half of it, draw two arcs which intersect each other at P and Q.

(iv) Join PQ. Then PQ is the perpendicular bisector of AB¯¯¯¯¯¯¯$\overline{\text{AB}}$.

We observe that this perpendicular bisector of AB¯¯¯¯¯¯¯$\overline{\text{AB}}$ passes through the centre C of the circle.

Question 8.Draw a circle of radius 4 cm. Draw any two of its chords. Construct the perpendicular bisectors of these chords. Where do they meet?

Answer: Steps of construction:

(i) Draw the circle with O and radius 4 cm.

(ii) Draw any two chords AB¯¯¯¯¯¯¯$\overline{\text{AB}}$ and CD¯¯¯¯¯¯¯$\overline{\text{CD}}$ in this circle.

(iii) Taking A and B as centers and radius more than half AB, draw two arcs which intersect each other at E and F.

(iv) Join EF. Thus EF is the perpendicular bisector of chord  AB¯¯¯¯¯¯¯$\overline{\text{AB}}$ .

(v) Similarly draw GH the perpendicular bisector of chord CD¯¯¯¯¯¯¯$\overline{\text{CD}}$.

These two perpendicular bisectors meet at O, the centre of the circle.

Question 9.Draw any angle with vertex O. Take a point A on one of its arms and B on another such that OA = OB. Draw the perpendicular bisectors of OA¯¯¯¯¯¯¯¯$\overline{\text{OA}}$ and OB¯¯¯¯¯¯¯.$\overline{\text{OB}}.$ Let them meet at P. Is PA = PB?

Answer: Steps of construction:

(i) Draw any angle with vertex O.

(ii) Take a point A on one of its arms and B on another such that

(iii) Draw perpendicular bisector of OA¯¯¯¯¯¯¯¯$\overline{\text{OA}}$ and OB¯¯¯¯¯¯¯.$\overline{\text{OB}}.$

(iv) Let them meet at P. Join PA and PB.

With the help of divider, we check that PA¯¯¯¯¯¯¯=PB¯¯¯¯¯¯¯.