1. **Problem:** Construct triangle ABC with $AB=5$ cm, $\angle ABC=75^\circ$, and circumradius $R=3.5$ cm using ruler and compass. Step 1: Draw segment $AB=5$ cm. Step 2: At point $B$, construct an angle of $75^\circ$ using compass and ruler. Step 3: Since circumradius $R=3.5$ cm, draw a circle with center $O$ (circumcenter) such that $OA=OB=OC=3.5$ cm. Step 4: Locate point $C$ on the ray from $B$ forming $75^\circ$ with $AB$ such that $C$ lies on the circumcircle. Step 5: Connect $AC$ to complete triangle $ABC$. 2. **Problem:** Construct triangle ABC with $BC=6.4$ cm, $CA=5.8$ cm, $\angle ABC=60^\circ$ and draw its incircle. Step 1: Draw segment $BC=6.4$ cm. Step 2: At point $B$, construct an angle of $60^\circ$. Step 3: Using compass, mark point $A$ on the ray from $B$ such that $CA=5.8$ cm. Step 4: Connect $AB$. Step 5: To draw incircle, construct angle bisectors of two angles; their intersection is the incenter $I$. Step 6: Draw a circle centered at $I$ tangent to any side; this is the incircle. Step 7: Measure radius of incircle using ruler. 3. (i) **Problem:** Construct triangle ABC with $AB=4$ cm, $BC=6$ cm, $\angle ABC=90^\circ$. Step 1: Draw segment $BC=6$ cm. Step 2: At point $B$, construct a right angle ($90^\circ$). Step 3: Mark point $A$ on the perpendicular ray from $B$ such that $AB=4$ cm. Step 4: Connect $AC$. (ii) **Problem:** Construct circumcircle passing through points $A,B,C$ and mark center $O$. Step 1: Construct perpendicular bisectors of at least two sides. Step 2: Their intersection is circumcenter $O$. Step 3: Draw circle centered at $O$ passing through $A,B,C$. 4. **Problem:** Construct triangle ABC with $BC=6$ cm, $AB=5.5$ cm, $\angle ABC=120^\circ$. Step 1: Draw segment $BC=6$ cm. Step 2: At $B$, construct $120^\circ$ angle. Step 3: Mark point $A$ on the ray such that $AB=5.5$ cm. Step 4: Connect $AC$. (i) Construct circumcircle by finding circumcenter $O$ via perpendicular bisectors and drawing circle through $A,B,C$. (ii) To draw cyclic quadrilateral $ABCD$ with $D$ equidistant from $B$ and $C$: - Step 1: Draw circle circumscribing $ABC$. - Step 2: Find point $D$ on the circle such that $DB=DC$ (i.e., $D$ lies on the perpendicular bisector of $BC$). - Step 3: Connect $AD$ to complete quadrilateral. 5. **Problem:** Construct triangle ABC with $BC=6.5$ cm, $AB=5.5$ cm, $AC=5$ cm and its incircle. Step 1: Draw segment $BC=6.5$ cm. Step 2: Using compass, draw arcs from $B$ and $C$ with radii $AB=5.5$ cm and $AC=5$ cm respectively; their intersection is $A$. Step 3: Connect $AB$ and $AC$. Step 4: Construct angle bisectors to find incenter $I$. Step 5: Draw incircle centered at $I$ tangent to sides. Step 6: Measure radius of incircle. 6. **Problem:** Construct a regular hexagon with side $5$ cm and circumscribing circle. Step 1: Draw circle with radius $5$ cm. Step 2: Mark a point on circle as first vertex. Step 3: Using compass set to $5$ cm, step around circle marking six points. Step 4: Connect consecutive points to form hexagon. Step 5: The original circle is the circumscribing circle. Final notes: All constructions use ruler and compass steps as described.