1. **Stating the problem:** Construct triangle $\triangle ABC$ and draw its circumcircle for the given cases. (i) $mAB = 5.5$ cm, $mAC = 6$ cm, and $m\angle A = 50^\circ$ (ii) $mAB = 6$ cm, $mBC = 4.5$ cm, and $mAC = 5$ cm 2. **Important concepts:** - The circumcircle of a triangle is the unique circle passing through all three vertices. - To construct the circumcircle, first construct the triangle accurately. - The circumcenter is the point where the perpendicular bisectors of the sides intersect. 3. **Construction steps for (i):** - Draw segment $AB$ of length 5.5 cm. - At point $A$, construct an angle of $50^\circ$. - From $A$ along this angle, mark point $C$ such that $AC = 6$ cm. - Connect points $B$ and $C$ to complete $\triangle ABC$. - Construct perpendicular bisectors of at least two sides (e.g., $AB$ and $AC$). - The intersection of these bisectors is the circumcenter $O$. - With center $O$ and radius $OA$ (or $OB$, or $OC$), draw the circumcircle. 4. **Construction steps for (ii):** - Draw segment $AB$ of length 6 cm. - Draw segment $BC$ of length 4.5 cm. - Using compass, from $A$ draw an arc with radius 5 cm. - Using compass, from $C$ draw an arc with radius 6 cm (length $AB$ is 6 cm, so $AC$ is 5 cm, $BC$ is 4.5 cm; here $AC=5$ cm). - The intersection of these arcs is point $C$. - Connect $A$ to $C$ and $B$ to $C$ to complete $\triangle ABC$. - Construct perpendicular bisectors of two sides. - Their intersection is the circumcenter $O$. - Draw the circumcircle centered at $O$ passing through $A$, $B$, and $C$. 5. **Summary:** - Use ruler and protractor for lengths and angles. - Use compass for arcs and circumcircle. - The circumcircle passes through all vertices of the triangle. This completes the construction and circumcircle drawing for both cases.