**Practical Geometry**

**Question 1.Draw of length 7.3 cm and find its axis of symmetry.**

**Solution:***Steps of Construction:*

- Draw a line segment AB = 7.3 cm.
- With centre A and radius > AB, draw arcs one on each side of AB.
- With centre B and the same radius as before, draw arcs cutting the previously drawn arcs at C and D respectively.
- Join CD intersecting AB at M. Then M bisects the line segment AB as shown.

The line segment so obtained is the required axis of symmetry.

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

**Solution:****Steps of Construction:**

- Draw a line segment AB = 9.5 cm.
- With centre A and radius > AB, draw arcs one on each side of AB. . A
- With centre B and the same radius as before, draw arcs cutting the previously drawn arcs at C and D respectively.
- Join CD. Then the line segment CD is the required perpendicular bisector of AB.

**Question 3.Draw the perpendicular bisector of 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 , what can you say about the lengths MX and XY?

**Solution: Steps of Construction:**

- Draw a line segment XY =10.3 cm.
- With centre X and radius > XY, draw arcs one on each side of XY.
- With centre Y and the same radius as before, draw arcs cutting the previously drawn arcs at A and B respectively.
- oin .AB intersecting XY at M. Then, AB is the perpendicular bisector of XY.
**(a)**Mark any point P on AB, the perpendicular bisector. On measuring, we find that PX = PY.**(b)**Since M is the mid-point of the segment XY. Therefore,

MX = XY = x 10.3 cm

= 5.15 cm

**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:**

- Draw a line segment AB = 12.8 cm.
- With centre A and radius > AB, draw arcs one on each side of AB.
- With centre B and the same radius as before, draw arcs cutting the previously drawn arcs at C and D respectively.
- Join CD intersecting AB at M.
- Further find the mid-points M
_{1}and M_{2}of AM and MB respectively proceeding in the same way qs-before.

∴ AM_{1}= M_{1}M = MM_{2}= M_{2}B, On measuring, we find that each part = 3.2 cm.

**Question 5.With of length 6.1 cm as diameter draw a circle.**

**Solution:****Steps of Construction:**

- Draw a line segment PQ = 6.1 cm.
- Bisect the segment PQ by drawing its perpendicular bisector. Let M be its mid-point.
- M as centre and radius = MP draw a circle.

The circle so obtained is the required circle.

**Question 6.Draw a circle with centre C and radius 3.4 cm. Draw any chord . Construct the perpendicular bisector of and examine if it passes through C.**

**Solution:****Steps of Construction:**

- Mark a point C on the plane of a paper.
- With C as centre and radius 3.4 cm, draw a circle.
- Let AB be any chord to this circle.
- Draw PQ, the perpendicular bisector of chord AB.

Clearly, this perpendicular bisector passes through C, the centre of the circle.

**Question 7.Repeat Question 6, if AB happens to he a diameter.**

**Solution:**

If AB happens to be a diameter then C will be the mid-point of the diameter 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:**

- Mark a point O on the plane of paper.
- With O as centre, draw a circle of radius 4 cm.
- Let AB and CD be any two chords of this circle.
- Draw PQ and RS the perpendicular bisectors of chords AB and CD respectively.

Clearly, these perpendicular bisectors pass through 0, 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 and . Let them meet at P. Is PA = PB?**

**Solution:****Steps of Construction:**

- Draw any angle XOY.
- Take a point A on OX and a point B on OY such that OA = OB.
- Draw CD and EF, the perpendicular bisectors of OA and OB respectively. Let them meet at P.

On measuring, we find that PA = PB.