Problem-solving
8.16 Constructions
Show all working. Partial marks are given for method.
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1**Constructing an equilateral triangle.** Use ruler and compass to construct an equilateral triangle on the segment AB = 8 cm. Then: (a) Verify the construction works (all sides equal). (b) Find the height from one vertex to the opposite side. (c) Calculate the area.
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2**Perpendicular bisector.** Construct the perpendicular bisector of a 12 cm segment AB. (a) Describe the construction. (b) Verify the midpoint is 6 cm from each end. (c) Justify why the bisector is the locus of points equidistant from A and B.
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3**Hexagonal patio plan.** Design a regular hexagonal patio with side 3 m using compass-and-ruler-only methods on a 1:50 scale plan. (a) State the side length on the plan. (b) Describe the construction. (c) Find the area of the patio (real).
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4**Constructing 75°.** Construct an angle of 75° using only ruler and compass. (a) Describe two different methods. (b) Verify with a protractor (in principle).
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5**Seed of life.** Starting with one central circle of radius 4 cm, construct six interlocking circles forming the "seed of life". (a) Describe the procedure step-by-step. (b) Find the radius of the surrounding hexagon's circumcircle. (c) Why does the construction close exactly after 6 circles?
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6**Constructing a square from a circle.** A circle has radius 5 cm. Construct a square such that all four corners touch the circle. (a) Find the side length of the inscribed square. (b) Find the area of the square. (c) What fraction of the circle is the square?
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7**Angle bisection.** Bisect a 100° angle using compass and straightedge. (a) Describe the procedure. (b) Find the resulting half-angle. (c) Use the bisection again to find a 25° angle.
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8**Constructing a regular octagon.** Inscribe a regular octagon in a circle of radius 6 cm. (a) Outline the construction (starting from a square). (b) Find the side length of the octagon. (c) Find the area.
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9**Triangle from three lengths.** Construct a triangle with sides 4 cm, 7 cm, 9 cm. (a) Verify the triangle inequality. (b) Describe the construction. (c) Find the height from the 7 cm side to the opposite vertex.
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10**Constructing a parallel line.** Through point P (not on line $\ell$), construct a line parallel to $\ell$ using only compass and ruler. (a) Describe the method. (b) Justify why the new line is parallel. (c) Use this to construct a parallelogram with side 6 cm and adjacent side 4 cm.
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11**Apothem and circumradius.** A regular hexagon has side 6 cm. (a) Construct the hexagon. (b) Find the inscribed-circle radius (apothem). (c) Find the circumscribed-circle radius. (d) Compute the ratio of the two radii.
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12**Combining constructions.** Construct (i) an equilateral triangle of side 6 cm, (ii) inscribe its circumscribed circle, and (iii) inscribe a regular hexagon in the same circle. (a) Describe each step. (b) Find the side length of the hexagon. (c) Compare the area of the triangle and the hexagon.
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