To construct a regular octagon given the diagonal, i. o. within ■ given circle (Fig. 2/29)
1. Draw the circle and insert a diameter AE.
2. Construct another diagonal CO, perpendicular to the first diagonal.
3. Bisect the four quadrants thus produced to cut the circle in B. D. F, andH.
ABCDEFGH is the required octagon.
To construct a regular octagon given the diameter, i.e. within a given square (Fig. 2/30)
1. Construct a square PORS. length of side equal to the diameter.
2. Draw the diagonals SQ and PR to intersect m T.
3. With centres P. Q. R and S draw four arcs, radius PT (a QT = RT = ST) to cut the square in A, B. C, D. E. F.GandH.
ABCDEFGH is the required octagon
To construct any g i van polygon, given the length of aside
There ere three fairly simple ways of constructing a regular polygon. Two methods require e simple calculation and the third requires very careful construction H it is to be exact. All three methods are shown. The constructions work for any polygon, and a heptagon (seven sides) has been chosen to illustrate them. Method 1 (Fig. 2/31)
1. Oraw a line AB equal in length to one of the sides end produce AB to P.
2. Calculate the exterior angle of the polygon by dividing 360® by the number of sides. In this case the extenor angle is 360*/7 - 51 377.
3. Draw the exterior angle PBC so that BC-AB.
4 Bisect AB and BC to intersect in 0
5 Draw a circle, centre 0 and radius OA (- OB - OC). 6. Step off the sides of the figure from C to D. D to E. etc. ABCDEFG is the required heptagon.
1. Drew a line A8 equal in length to one o! the side».
2. From A erect e semi-circle, radius AB to meet BA produced in P.
3. Divide the semi-circle into the same number of equal parts as the proposed polygon has sides. This may be done by trial and enor or by calculation (18077 -25 5*/7 for each arc).
4. Draw a line from A to point 2 (for ALL polygons). This forms a second side to the polygon.
5. Bisect AB and A2 to intersect in 0.
6. With centre O draw a circle, radius OB (- OA - 02).
7. Step off the sides of the figure from B to C. C to D etc
ABCDEFG is the required septagon
1. Draw a line GA equal in length to one of the sides
3. From A construct an angle of 45** to intersect the bisector at point 4.
4. From G construct an angle of 60p to intersect the bisector at point 6.
Point 4 is the centre of a circle containing a square. Point 5 is the centre of a circle containing a pentagon. Point 6 is the centre of a circle containing a hexagon By marking off points at similar distances the centres of circles containing any regular polygon can be obtained.
6. Mark off point 7 so that 6 to 7 - 5 to 6 ( — 4 to 5).
7. With centre at point 7 draw a circle, redius 7 to A (- 7 toG).
8. Step off the sides of the figure from A to B. B to C. etc. ABCDEFG is the required heptagon.
To construct a regular polygon given e diagonal.
1. Draw the given circle and insert a diameter AM.
2. Divide the diameter into the same number of divisions as the polygon has sides.
3. With centre M draw an arc. radius MA With centre A draw another arc of the same radius to intersect the first arc in N.
4. Draw N2 and produce to intersect the circle in B (for any polygon).
5. AB is the first side of the polygon. Step out the other sides 8C. CO. etc.
ABCDE is the required polygon.
To construct • regular polygon given e diameter (Fig. 2/36)
1. Draws line MN.
2. From some point A on the line draw a semi-circle of any convenient radius.
3. Divide the semi-circle into the same number of equal sectors as the polygon has sides (in this case 9, i.e. 208 intervals).
5. If the polygon has an even number of sides, there is only one diameter passing through A. In this case, bisect the known diameter to give centre 0. If, as in this case, there are two diameters passing through A (there can never be more than two), then bisect both diameters to intersect in 0.
6. With centre O and radius OA draw a circle to intersect the radial lines in C, D. E, F. G and H.
ABCDEFG H J is the required polygon.
The constructions shown above are by no means all the constructions that you may be required to do. but they are representative of the type that you may meet.
If your geometry needs a little extra practice, it is well worth while proving these constructions by Euclidean proofs. A knowledge of some geometric theorems is needed when answering many of the questions shown below, and proving the above constructions will make sure that you are familiar with them.
Exercises 2
1. Construct an equilateral triangle with sides 60 mm long.
2. Construct an isosceles triangle that has a perimeter of 135 mm and an altitude of 65 mm.
3. Construct a triangle with base angles 60* and 45* and an altitude of 76 mm.
4. Construct a triangle with a base of 55 mm. an altitude of 62 mm and a vertical angle of 37}*.
6. Construct a triangle with a perimeter measunng 160mm and sides in the ratio 3:5:6.
6. Construct a triangle with a perimeter of 170 mm and sides in the ratio 7:3:5.
7. Construct a triangle given that the perimeter is 115 mm. the altitude is 40 mm and the vertical angle is 45*.
8. Construct a triangle with a base measuring 62 mm. an altitude of 50 mm and a vertical angle of 60*. Now drew a similar triangle with a perimeter of 250 mm
9. Construct a tnangle with a perimeter of 125 mm whose sides are in the ratio 2:4:5. Now draw a similar triangle whose perimeter is 170 mm.
10. Construct a square of side 50 mm. Find the mid-point of each side by construction and join up the points with straight lines to produce a second square
11. Construct a square whose diagonal is 68 mm.
12. Construct a square whose diagonal is 85 mm.
13. Construct a parallelogram given two sides 42 mm and 90 mm long, and the angle between them 67".
14. Construct a rectangle which has a diagonal 55 mm long end one side 35 mm long.
15. Construct a rhombus if the diagonal is 75 mm long and one side is 44 mm long.
16. Construct a trapezium given that the parallel sides are 50 mm and 80 mm long and are 45 mm apart.
17. Construct a regular hexagon. 45 mm side.
18. Construct a regular hexagon if the diameter is 75 mm.
19. Construct a regular hexagon within an 80 mm diameter circle. The corners of the hexagon must all lie on the circumference of the circle
20. Construct a square, side 100 mm. Within the square, construct a regular octagon. Four alternate sides of the octagon must lie on the sides of the square
21. Construct the following regular polygons:
a pentagon, side 65 mm, a heptagon, side 55 mm. a nonegon. side 45 mm. a decagon, side 36 mm.
22. Construct a regular pentagon, diameter 82 mm.
23. Construct a regular heptagon within a circle, radius 60 mm. The corners of the heptagon must lie on the circumference of the circle.
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