Gujarat Board GSEB Textbook Solutions Class 6 Maths Chapter 14 Practical Geometry Ex 14.5 Textbook Questions and Answers.
Gujarat Board Textbook Solutions Class 6 Maths Chapter 14 Practical Geometry Ex 14.5
Question 1.
Draw \overline{\mathrm{AB}} of length 7.3 cm and find its axis of symmetry.
Note: The perpendicular bisector of a line segment is its axis of symmetry.
Solution:
Steps of construction:
Step I: Draw line segment \overline{\mathrm{AB}} = 7.3 cm.
Step II: With centres A and B and radius more than half of AB, draw two arcs that intersect each other at P and Q.
Step III: Join P and Q.
Thus, PQ is the axis of symmetry of \overline{\mathrm{AB}}.
Question 2.
Draw a line segment of length 9.5 cm and construct its perpendicular bisector.
Solution:
Steps of construction:
Step I: Draw a line segment AB = 9.5 cm.
Step II: With A and B as centres and radius more than half of \overline{\mathrm{AB}}, draw two arcs on either side of AB which intersects each other at P and Q.
Step III: Join P and Q.
Thus, PQ is the required perpendicular bisector of \overline{\mathrm{AB}}.
Question 3.
Draw the perpendicular bisector of \overline{\mathrm{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 \overline{\mathrm{XY}}, what can you say about the lengths MX and XY?
Solution:
Steps of construction:
Step I: Draw a line segment \overline{\mathrm{XY}} = 10.3 cm
Step II: With X and Y as centres and radius more than half of XY, draw two arcs on either side of XY which intersect each other at A and B.
Step III: join A and B.
Thus, AB is ⊥ \overline{\mathrm{XY}}
Step IV: Mark a point ‘P’ on AB and join PX and PY.
(a) On measuring \overline{\mathrm{PX}} and \overline{\mathrm{PY}} (using a divider) we get that \overline{\mathrm{PX}} = \overline{\mathrm{PY}} .
(b) The mid-point of XY is M. On measuring,
we have that \overline{\mathrm{XM}} = \overline{\mathrm{MY}} = \frac { 1 }{ 2 } XY.
Question 4.
Draw a line segment of length 12.8 cm. Using compasses, divide it into four equal parts. Verify by actual measurement.
Solution:
Steps of construction:
Step I: Draw a line segment \overline{\mathrm{AB}} = 12.8 cm.
Step II: Draw perpendicular bisector of AB, which meets \overline{\mathrm{AB}} at O. (i.e. O is the mid-point of \overline{\mathrm{AB}})
Step III: Draw perpendicular bisector of \overline{\mathrm{AO}}, which meet \overline{\mathrm{AB}} at P. {i.e. P is mid-point of \overline{\mathrm{AO}})
Step IV: Now, draw perpendicular bisector of \overline{\mathrm{BO}}, which meets AB at Q. (i.e. Q is the mid-point of OB)
Step V: The line segment is divided into 4 equal parts by the points P, 0 and Q.
Step VI: By actual measurement, we have:
\overline{\mathrm{AP}} = \overline{\mathrm{PO}} = \overline{\mathrm{OQ}} = \overline{\mathrm{QB}} = 3.2 cm
Question 5.
With \overline{\mathrm{PQ}} of length 6.1 cm as diameter, draw a circle.
Solution:
Steps of construction:
Step I: Draw a line segment \overline{\mathrm{XY}} =6.1 cm.
Step II: Draw the perpendicular bisector of PQ which meets \overline{\mathrm{XY}} at O. (i.e. O is the mid-point of PQ).
Step III: With centre O and \overline{\mathrm{OP}} or \overline{\mathrm{OQ}} as radius, draw a circle passing through P and Q.
The circle having \overline{\mathrm{XY}} as the diameter is the required circle.
Question 6.
Draw a circle with centre C and radius 3.4 cm. Draw any chord \overline{\mathrm{AB}}.Construct the perpendicular bisector of \overline{\mathrm{AB}} and examine if it passes through C.
Solution:
Steps of construction:
Step I: Mark a point C on a paper.
Step II: With centre ‘C’ and radius 3.4 cm, draw a circle.
Step III: Draw a chord \overline{\mathrm{AB}}.
Step IV: Draw perpendicular bisector of \overline{\mathrm{AB}}.
We find that l (the perpendicular bisector of \overline{\mathrm{AB}}) passes through the centre of the circle.
Question 7.
Repeat question 6, if \overline{\mathrm{AB}} happens to be a diameter.
Solution:
Steps of construction:
Step I: Mark a point ‘C’ on a paper.
Step II: With centre C and radius 3.4 cm, draw a circle.
Step III: Draw a diameter \overline{\mathrm{AB}} of the circle.
Step IV: Draw the perpendicular bisector ‘l’ of \overline{\mathrm{AB}}. We find that l passes through C and C is the mid-point of AB.
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?
Solution:
Steps of construction:
Step I: Mark a point O on a paper.
Step II: With centre O and radius 4 cm, draw a circle.
Step III: Draw two chords \overline{\mathrm{AB}} and \overline{\mathrm{CD}}.
Step IV: Draw perpendicular bisector ‘l’ of \overline{\mathrm{AB}}.
Step V: Draw perpendicular bisector ‘M’ of \overline{\mathrm{CD}}.
Step VI: Produce l and m to meet each other. We find that l and m 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 \overline{\mathrm{OA}} and \overline{\mathrm{OB}}. Let them meet at P. Is PA = PB?
Solution:
Steps of construction:
Step I: Mark a point O on a paper.
Step II: Draw an angle ∠XOY, having a vertex at O.
Step III: Mark a point A on \overrightarrow{\mathrm{OX}} and another point B on \overrightarrow{\mathrm{OY}}, such that \overline{\mathrm{OA}} = \overline{\mathrm{OB}}.
Step IV: Draw l, the perpendicular bisector of \overline{\mathrm{OB}} .
Step V: Draw m, the perpendicular bisector of \overline{\mathrm{OA}}
Step VI: Mark the intersecting points of l and m as P.
Step VII: Join \overline{\mathrm{PA}} and \overline{\mathrm{PB}}, measure them with the help of a divider.
Step VIII: On measuring, we find \overline{\mathrm{PA}} = \overline{\mathrm{PB}}