31.

Use ruler and compasses only for the following questions. All constructions lines and arcs must be clearly shown.

(i) Construct a $∆$ABC  in which  BC = 6.5 cm,  $\angle$ABC = 60⁰,  AB = 5 cm.

(ii) Construct the locus of points at a distance of  3.5 cm  from  A.

(iii) Construct the locus of points equidistant from  AC  and  BC.

(iv) Mark  2  points  X  and  Y  which are a distance of  3.5 cm  from  A  and also equidistant from  AC  and  BC. Measure  XY.

( i )  Steps of construction:

1. Draw  BC = 6.5 cm using a ruler.

2. With  B  as the centre and radius equal to approximately half of  BC, draw an arc that cuts the segment  BC  at  Q.

3. With  Q  as the centre, and same radius, cut the previous arc at  P.

4. Join  BP  and extend it.

5. With  B  as the centre and radius  5 cm, draw an arc that cuts the

arm  PB  to obtain point  A.

6. Join  AC  to obtain  ΔABC.

( ii )  Steps for construction:

1. With  A  as the centre and radius  3.5 cm, draw a circle.

2. The circumference of a circle is the required locus.

( iii ) Steps for construction:

1.With  C  as the centre and with radius of a length less than  CA or  BC,
draw an arc to cut the line segments  AC  and  BC  at  D  and  E  respectively.

2. With the same radius or a suitable radius and with  D  as the centre, draw an arc of a circle.

3. With the same radius and with  E  as the centre draw an arc such that
the two arcs intersect at  H.

4. Join  C  and  H.

5. CH  is the bisector of  $\angle$ACB and is the required locus.

( iv ) Steps for construction:

1.We known that the points at a distance of  3.5 cm  from  A  will surely lie on the circle with centre  A.

2. Also, the points on the angle bisector  CH  are the points equidistant from  AC  and  BC.

3. Mark  X  and  Y  which are at the intersection of the circle and the angle bisector  CH.

4. Measure  XY . XY  =  5 cm.