Construction of Triangle ABC and the Required Loci

Construct triangle ABC using a ruler and pair of compasses only, given that:

• |AB| = 8.4 cm
• |BC| = 6.5 cm
• ∠ABC = 30°

1. Construct triangle ABC

1. Draw a line segment BC = 6.5 cm.
2. At point B, construct a 60° angle using the equilateral-triangle construction: with centre B and any suitable radius, draw an arc; using the same radius from a point on the arc, mark the intersection to form a 60° ray.
3. Bisect the 60° angle to obtain a ray making 30° with BC.
4. With centre B and radius 8.4 cm, draw an arc cutting the 30° ray at A.
5. Join A to C. This completes triangle ABC.

2. Construct locus l₁

The locus l₁ of points equidistant from the lines AB and BC, and lying within ∠ABC, is the internal angle bisector of ∠ABC.

1. With centre B, draw an arc cutting BA and BC at points D and E.
2. With the same compass radius, draw arcs from D and E to meet at F inside ∠ABC.
3. Draw the line BF. This line is locus l₁.

3. Construct locus l₂

The locus l₂ of points equidistant from points B and C is the perpendicular bisector of BC.

1. Set the compass radius to more than half of BC.
2. With centres B and C, draw arcs above and below BC so that they intersect.
3. Join the two arc-intersection points. The resulting line is locus l₂.

Therefore:

• l₁ is the internal bisector of ∠ABC.
• l₂ is the perpendicular bisector of BC.
• I = l₁ ∩ l₂.