Stage 9 - Q1 - 2 marks
On plain paper draw a circle of radius 32 mm. Inscribe a square using ruler and compasses. Keep the construction arcs.
Hint
Perpendicular diameters provide the four vertices.
Stage 9 - Q2 - 2 marks
A circle of radius 34 mm is divided by two perpendicular diameters. Explain why joining their four endpoints makes a regular square.
Hint
Use equal central sectors and the four-sided angle sum.
Stage 9 - Q3 - 3 marks
Draw a circle of radius 36 mm. Keep the compass at that radius and step around the circle to inscribe a regular hexagon.
Hint
Do not change the compass opening between vertices.
Stage 9 - Q4 - 3 marks
A regular hexagon is inscribed in a circle of radius 38 mm. Find each central angle and each interior angle, and explain why they differ.
Hint
A central angle is at the centre; an interior angle is at a polygon vertex.
Stage 9 - Q5 - 4 marks
Draw a circle of radius 40 mm. Construct perpendicular diameters, bisect each right-angle sector and join the eight endpoints.
Hint
Bisect the central angles, not the polygon side lengths.
Stage 9 - Q6 - 4 marks
Draw a circle of radius 42 mm. Construct perpendicular diameters, bisect each right-angle sector and join the eight endpoints.
Hint
Bisect the central angles, not the polygon side lengths.
Stage 9 - Q7 - 5 marks
Start with four equal central sectors. Bisect every sector 3 times. How many equal sectors result, and what regular polygon is obtained by joining their circle endpoints?
Hint
Track the count and angle in parallel.
Stage 9 - Q8 - 5 marks
Use a circle of radius 47 mm to construct a regular dodecagon. Begin with a regular hexagon and explain the extra step.
Hint
Insert one new vertex in every original central sector.
Stage 9 - Q9 - 5 marks
In a circle of radius 48 mm, a regular inscribed polygon also has side 48 mm. Prove that it must be a hexagon.
Hint
Use the triangle made by the centre and two neighbouring vertices.
Stage 9 - Q10 - 3 marks
Start with an equilateral triangle inscribed in a circle. Using only central-angle bisection, construct a regular dodecagon. How many rounds of bisection per original sector and how many new vertices are required? Justify the count.
Hint
Count the points inserted inside each original gap; do not recount its two existing endpoints.