Loci — Analytical & Application Based Questions

Construct a triangle ABC such that AB = 7 cm, BC = 6 cm and CA = 5 cm (use ruler and compass to do so).

(a) Draw the locus of the points such that

(i) it is equidistant from BC and BA.

(ii) it is equidistant from points A and B.

(b) Mark P where the loci (i) and (ii) meet, measure and write length of PA.

Answer

We know that,

Locus of the points equidistant from two sides is the angle bisector between two sides.

Locus of the points equidistant from two points is the perpendicular bisector of the line joining two points.

Steps of construction :

1. Draw a line segment AB = 7 cm.

2. With A and B as center and radii 5 cm and 6 cm respectively draw arcs intersecting at C.

3. Join AC and BC.

4. Draw BD, angle bisector of B.

5. Draw XY perpendicular bisector of AB.

6. Mark point P, the intersection point of XY and BD.

7. Measure AP.

Hence, AP = 3.8 cm.

(Use a ruler and a compass for this question.)

(a) Construct the locus of a moving point which moves such that it keeps a fixed distance of 4.5 cm from a fixed-point O.

(b) Draw line segment AB of 6 cm where A and B are two points on the locus (a).

(c) Construct the locus of all points equidistant from A and B. Name the points of intersection of the loci (a) and (c) as P and Q respectively.

(d) Join PA. Find the locus of all points equidistant from AP and AB.

(e) Mark the point of intersection of the locus (a) and (d) as R. Measure and write down the length of AR.

Answer

We know that,

Locus of a point at a fixed distance from a fixed point is the circumference of the circle with fixed point as center and fixed distance as radius.

Steps of construction :

1. Mark point O.

2. With center O and radius = 4.5 cm draw a circle.

3. Use a ruler to mark point A.

4. From point A, measure 6 cm along a straight line and mark point B.

5. Draw a straight line connecting points A and B.
   We know that, locus of points equidistant from two points is the perpendicular bisector of the line joining the two points.

6. Draw perpendicular bisector of AB.

7. Mark the points P and Q where the perpendicular bisector intersects circle.
   We know that, locus of points equidistant from two lines is the angle bisector of the angle between the lines.

8. Join PA.

9. Draw AX, the angle bisector of ∠A.

10. Mark point R as the intersection point of AX on the circumference of the circle.

11. Measure AR.

Hence, AR = 4.8 cm.