Using a rular and compass only :
(i) Construct a triangle ABC with the following data :
AB = 3.5 cm, BC = 6 cm and `angleABC=120^(@)`.
(ii) In the same diagram, draw a circle with BC as diameter. Find a point P on the circunference of the circle which is equidistant from AB and BC.
(iii) Measure `angleBCP`.

Answer

### Step-by-Step Solution

#### (i) Construct Triangle ABC

1. **Draw Line Segment BC:**
- Use a ruler to draw a line segment BC measuring 6 cm. Label the endpoints as B and C.

2. **Construct Angle ABC:**
- At point B, use a compass to draw a semicircle with a radius greater than half of BC. The semicircle will intersect the line BC at two points.
- Mark the intersection points on the semicircle. Label one of the intersection points as D.
- Using the compass, measure the distance from B to D, and then draw an arc from D to create an angle of 120 degrees at B.

3. **Locate Point A:**
- Using the compass, measure 3.5 cm from point B along the ray you just drew (the ray forming the angle with BC). Mark this point as A.

4. **Join Points A and C:**
- Use the ruler to draw a line segment connecting points A and C. Triangle ABC is now constructed.

#### (ii) Draw Circle with BC as Diameter and Find Point P

1. **Draw the Perpendicular Bisector of BC:**
- Using a compass, place the pointer at point B and draw arcs above and below the line segment BC.
- Without changing the compass width, repeat this from point C. The two arcs will intersect at two points. Draw a line through these intersection points to create the perpendicular bisector. Label the intersection point as O.

2. **Draw Circle with Diameter BC:**
- Using O as the center and the distance OC as the radius, draw a circle. This circle will have BC as its diameter.

3. **Construct the Angle Bisector of Angle ABC:**
- To find point P, draw the angle bisector of angle ABC. Place the compass pointer at point A and draw an arc that intersects both lines AB and BC. Label these intersection points as E and F.
- Without changing the compass width, place the pointer at points E and F to create two arcs above the angle. Draw a line from B through the intersection of these arcs. This line is the angle bisector.

4. **Locate Point P:**
- The intersection of the angle bisector with the circle is the point P. Mark this point.

#### (iii) Measure Angle BCP

1. **Measure Angle BCP:**
- Use a protractor to measure the angle BCP. The angle should measure 30 degrees.