Classwork Series and Exercises {Basic Technology – JSS2}: QUADRILATERALS AND POLYGONS

JSS 2 SECOND TERM

Basic Technology

Topic: QUADRILATERALS AND POLYGONS

DEFINITIONS

A quadrilateral may be defined as a plane figure bounded by four straight sides. It also has four angles. Any two opposite angle points may be joined by a straight line termed the diagonal.

A quadrilateral with its opposite sides equal and parallel is called a parallelogram.

A parallelogram which has all its sides equal and each angle a right angle is called a square.

A rhombus is a parallelogram which has all its sides equal but no angle is a right angle.

A parallelogram which has its opposite sides equal and each angle a right angle, is called a rectangle.

A rhomboid has its opposite sides equal and parallel but no angle is a right angle.

 

(A) To construct a square upon a given side

i. Draw a line and a mark off AB equal to the side of the square.

ii. At A, erect a perpendicular AC making AC equal to AB.

iii. With centre C and a radius equal to AB, strike an arc. With centre B and the same radius, strike another arc to intersect the previous one at D.

iv. Join CD and BD to obtain the required square ABCD.

(B) To construct a rectangle given its length and breadth

i. Draw a line and mark off AB equal to the length of the rectangle.

ii. At A, erect a perpendicular AC making AC equal to the breadth of the triangle.

iii. With centre C and a radius equal to AB, strike an arc. With centre B and a radius equal to the AC, strike another arc to intersect the previous one at D.

iv. Join CD and BD to obtain the required rectangle ABCD.

(C) To construct a square given the length of its diagonal

i. Draw a horizontal line and a vertical line which intersect at O.

ii. With centre O and a radius equal to half the length of the given diagonal, cut the horizontal line at Band the vertical line at C and D.

iii. Join AD, DB, BC and CA, obtain the required square ADBC.

(D) To construct a rectangle given its diagonal and one side

i. Draw a line and mark off AB equal to the given diagonal

ii. Bisect AB at C, and with centre C draw a circle with AB as diameter.

iii. With centre A and a radius equal to the given side of the rectangle, cut the circle on any side of the AB at D. With centre B and the same radius, cut the circle on any side of AB at E.

iv. Join AE, EB, BD and DA to obtain the required rectangle AEBD

(E) To construct a rhombus given its side and a diagonal

i. Draw a line and mark off AB equal to the given diagonal.

ii. With centre A and radius equal to the side, strike arcs above and below AB.

iii. With centre B and the same radius, cut the previous arcs at C and D.

iv. Join AD,DB, BC and CA to obtain the required rhombus ADBC.

(F) To construct a regular rhombus given a diagonal and two sides

i. Draw AB equal to the given diagonal.

ii. With centre A and B and a radius equal to one of the sides, strikes arcs above and below AB respectively.

iii. With centre B and a radius equal to the other side, strike arcs to intersect the previous ones at C and D respectively.

iv. Join AD, DB, BC and CA to obtain the required rhomboid ADBC.

v. With centre C and a radius equal to AB, strike an arc. With center B and the same radius, strike another arc to intersect the previous one at D.

Polygons

A polygon is a plane figure formed by joining three or more straight sides. A polygon is said to be regular if all its sides are equal and its angles are equal.

A pentagon is a polygon with five sides.

A hexagon is a polygon with six sides.

A heptagon has seven sides.

An octagon has eight sides.

A decagon has ten sides.

(A) To construct a regular a Regular Hexagon given its side

(I) Using 60⁰ set square

i. Draw a horizontal line and mark off AB equal to the side of the hexagon.

 

ii. Through A, draw a line at 60⁰ and mark off AC equal to AB.

iii. Through B, draw a line at 60⁰ parallel to BD and mark off BD equal to AB.

iv. Through C, draw a line of 60⁰ parallel to BD and mark off CE equal to AB.

v. Through D, draw a line of 60⁰ parallel to AC and mark off DF equal to AB.

vi. Join EF to complete the hexagon

(II) Using a pair of Compass

This method is best remembered as the constant Radius Rule.

i. Draw a circle whose radius is equal to the side of the hexagon. Draw the horizontal diameter AB.

ii. With centre A and the same radius, cut the circle above AB at E and below AB at F.

iii. With centre B and the same radius, cut the circle above AB at E and below AB at F.

iv. Join AD, DF, FB, BE, EC, and CA to obtain a hexagon.

Note: This procedure is required to draw a regular hexagon given the distance across corners. The diameter of the circle is equal to the distance across corners.

(B) To construct a regular hexagon given the distance across flats

i. Draw a circle whose diameter is equal to the distance across flats. Draw the vertical diameter AB.

ii. Draw diameter CD and EF at 30⁰.

iii. Through A and B, draw horizontal tangents.

iv. Through C, D, E, F, in turn, draw tangents at 60⁰. The figure that is formed by the intersection of the tangents is the required hexagon.

Hexagon

Note: This is the procedure when it is required to describe a regular hexagon about a given circle.

(C) To construct a regular octagon given its side

i. Through C and D, draw vertical lines and mark off CE and DF equal to AB.

ii. Through E and F, draw lines at 45⁰ and mark off EG and FH equal to AB.

iii. Join GH to complete the octagon.

iv. Through C and D, draw vertical lines and mark off CE and DF equal to AB.

v. Through E and F, draw lines at 45⁰ and mark off EG and FH equal to AB.

vi. Join GH to complete the octagon.

(D) To construct a regular octagon given the distance across flats

i. Draw a circle whose diameter is equal to the distance across flats. Draw horizontal diameter AB and vertical diameter CD.

ii. Draw diameters EF and GH at 45⁰.

iii. Draw vertical tangents through A and B and horizontal tangents through C and D.

iv. Through E, F, G H in turn draw tangents at 45⁰. The figure formed by the intersection of the tangents is the required octagon.

Note: This is the procedure when it is required to describe a regular about a given circle.

(E) General methods for constructing a regular polygon on a given base

(a) The ‘External – 360⁰/N Rule’

i. Obtain the external angle of the required polygon by dividing 360⁰ by the number of sides (N) of the polygon i.e. external angle = 360⁰/N.

ii. Draw a horizontal line and mark off AB equal to the given base.

iii. Through A, draw a line at 360⁰/N and mark off a length equal to AB. Also N at B, draw a line at 360⁰/N and mark off a length to AB.

iv. Continue the process until you have obtained the polygon N side where N = 5, 6, 7, 8, 9, 10, ….

Suppose that N = 5, then external angle = 360⁰/N = 72⁰. The pentagon will be obtained by drawing at 72⁰(b).

(b) The ‘Two-Triangle Rule’

i. Draw a horizontal line and mark off AB equal to the given base.

ii. Bisect AB and produce its bisector as long as it is convenient.

iii. On AB as base, draw an isosceles triangle with base angle 45⁰ and an equilateral triangle so that the apexes of the two triangles lie on the bisector of AB. Denote the apex of the isosceles triangle as d, and that of the equilateral triangle as f.

iv. Bisect fd to obtain point e.

v. Along the bisector of AB, from the point f, step off length de (or ef) to obtain points g, h, I, j, etc. The points d, e, f, g, I, j are the centres of the circumscribing circles for a square, regular pentagon, hexagon, heptagon, octagon, nonagon and decagon respectively.

vi. suppose you want to draw a polygon of 9 sides (nonagon). With centre I and radius I A (or iB) draw a circle. Take length AB and step it off on the circle to obtain the points, C, D, E, F, G, H, I.

Join the points to obtain the required regular nonagon.

(Observed that d = 4; e = 5; f= 6; g = 7; h = 8; i = 9; j = 10.)

(F) The General Method for Describing a Regular about a given Circle

The method is best remembered as the ‘Centre – 360⁰/N Rule’.

i. Obtain the angle included by any two normals at the centre of the given circle by dividing 360⁰/N by the number of sides N of the required described polygon, i.e. angle at centre = 360⁰/N.

ii. Draw the given circle with centre O and draw a vertical radius OA.

iii. Use a protractor to set out angles of 360⁰/N and draw radii of OB, OC, OD etc., until you have got N radii.

iv. Through the points A, B, C, D, etc., draw tangents to obtain the required polygon. Suppose it is required to draw an octagon.

360⁰/N = 360⁰/8 = 45⁰.