To construct the required geometrical figures and measurements, follow these steps carefully:

Step 1: Construct Triangle ABC

1. Draw Line Segment AB = 7 cm:

   • Using a ruler, draw a straight line segment AB with a length of 7 cm.

2. Construct Angle 105° at Point A:

   • Place the compass on point A, draw an arc across the line AB.
   • Keep the same compass width, place the compass point on where the arc cuts AB, and make another arc that cuts the first arc.
   • From the intersection of the arcs, draw an arc across AB using the compass.
   • Without changing the compass width, move the compass to where the second arc cuts the first arc, and draw another arc to intersect the previous one.
   • Using a ruler, draw a line from point A through the intersection of the arcs to create a 105° angle.

3. Draw Line Segment AC = 8 cm:

   • Place the compass point on point A, and set the compass to 8 cm (AC's length).
   • Draw an arc to intersect the line at 105° from AB at a point which we'll call C.

4. Draw Line Segment BC:

   • Finally, connect point B to point C with a straight line to complete triangle ABC.

Step 2: Construct X, the Locus of Points 6 cm from C

1. Draw a Circle with Radius 6 cm Centered at C:
   • Place the compass point on point C and set the compass to 6 cm.
   • Draw a full circle around point C. This circle represents the locus of points 6 cm away from point C.

Step 3: Construct Y, the Locus of Points Equidistant from Line AB and Line BC

1. Construct the Angle Bisector of ∠ABC:

   • To find the locus of points equidistant from AB and BC, you need to bisect angle ABC.
   • Place the compass on point B, draw arcs across both AB and BC.
   • Without changing the compass width, draw two arcs from the points where the first arcs intersected AB and BC.
   • The point where these arcs intersect is a point on the angle bisector. Draw the angle bisector from point B through this intersection point.

2. Extend the Angle Bisector:

   • Extend the angle bisector until it intersects with the circle (locus X) at two points, which we'll call P and R.

Step 4: Measure Line Segment BC

1. Measure BC:
   • Use a ruler to measure the length of the line segment BC.

Step 5: Measure Line Segment PR

1. Measure PR:
   • Use a ruler to measure the length of the line segment PR, which is the distance between points P and R.

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Results:

• Length of BC: (Use the ruler to measure this and note down the value.)
• Length of PR: (Use the ruler to measure this and note down the value.)

Would you like more details on any of these steps or have any questions?

Follow-up Questions:

1. How can you verify that the angle you constructed is exactly 105°?
2. What are the possible sources of error in the construction, and how can they be minimized?
3. How would you construct a perpendicular bisector of a line segment?
4. Can you explain why the locus of points equidistant from two lines is the angle bisector?
5. What would happen if the angle between AB and BC was 90°? How would that change the construction?
6. How would you construct the locus of points that are 4 cm away from a given point instead of 6 cm?
7. How does the construction change if triangle ABC is obtuse or acute?
8. What is the significance of the points P and R in geometric constructions?

Tip: Always double-check your compass width and angles during construction to ensure accuracy.