Hello, and welcome to your mathematics lesson today. We are going to explore the fascinating world of constructing polygons using only a ruler and compass. This is Chapter 15, and by the end of this lesson, you will know how to construct quadrilaterals, parallelograms, trapeziums, rectangles, rhombuses, squares, and even regular hexagons. So, let us begin.
Before we start any construction, here is a golden rule. Always draw a rough free-hand sketch first. This helps you visualize the shape and plan your steps. Remember, to construct a quadrilateral means to locate its four vertices precisely.
Let us begin with quadrilaterals. There are several ways to construct them depending on what information you are given.
First, suppose you know all four sides and one angle. Imagine quadrilateral A B C D, where A B equals 3.5 centimetres, B C equals 4.0 centimetres, C D equals 5.0 centimetres, D A equals 4.0 centimetres, and angle B equals 45 degrees. Here is how you proceed. Start by drawing B C equal to 4.0 centimetres. Through point B, draw a ray B P making a 45-degree angle with B C. From this ray, cut off B A equal to 3.5 centimetres. Now, with A as centre and radius 4 centimetres, draw an arc. With C as centre and radius 5 centimetres, draw another arc intersecting the first at point D. Join A D and C D. You have now constructed quadrilateral A B C D.
Second, suppose you know three sides and two consecutive angles. Let us say A B equals 4.0 centimetres, B C equals 4.5 centimetres, C D equals 4.7 centimetres, angle B equals 60 degrees, and angle A equals 120 degrees. Begin by drawing B C equal to 4.5 centimetres. Construct angle M B C equal to 60 degrees, then cut off B A equal to 4.0 centimetres from this ray. At A, draw a ray making 120 degrees with A B. With C as centre and radius 4.7 centimetres, draw an arc cutting this ray at D. Join C D, and your quadrilateral is complete.
Third, when you know four sides and one diagonal, the strategy changes. You first construct triangle A B C using the given information, then construct triangle A D C using the remaining sides. This divides the problem into two manageable triangle constructions.
Fourth, when given three sides and two diagonals, you again use the triangle method. Construct triangle A B C first, then triangle A B D, and finally join C to D. This elegant approach of breaking quadrilaterals into triangles is a powerful technique you will use repeatedly.
Now, let us move to parallelograms. These have special properties that make construction easier. Opposite sides are equal, and diagonals bisect each other.
When given two consecutive sides and the included angle, say A B equals 3.0 centimetres, B C equals 4.0 centimetres, and angle B equals 60 degrees, remember that D C must equal A B, and A D must equal B C. Construct triangle A B C first, then triangle A D C.
Here is a beautiful construction using diagonal properties. When given one side and both diagonals, say B C equals 4.5 centimetres, diagonal A C equals 5.6 centimetres, and diagonal B D equals 5.0 centimetres. Since diagonals bisect each other, let O be their intersection. Then O B equals half of B D, which is 2.5 centimetres, and O C equals half of A C, which is 2.8 centimetres. Construct triangle O B C with these lengths, then extend B O to D and C O to A, making O D equal to O B and O A equal to O C. Join the vertices to complete your parallelogram.
Sometimes measurements do not work out neatly. If half a diagonal gives a decimal like 2.25 centimetres, which your scale cannot measure precisely, simply start with the other diagonal instead. This flexibility is part of good geometric thinking.
When given two adjacent sides and the height, you use a different approach. Draw the base, erect a perpendicular, mark the height, draw a parallel line at that height, then use your compass to locate the remaining vertices from the ends of the base.
Next, the trapezium. Suppose A D is parallel to B C, with A D equal to 3.0 centimetres, A B equal to 2.5 centimetres, B C equal to 5.0 centimetres, and C D equal to 2.8 centimetres. Here is the clever step. From B C, cut off B E equal to A D, which is 3.0 centimetres. This creates triangle D E C where D E equals A B, which is 2.5 centimetres. Construct this triangle, then locate A by drawing arcs from B and D with appropriate radii. Join A B and A D to complete the trapezium.
Rectangles are special parallelograms with all angles equal to 90 degrees. When given adjacent sides, construct right-angled triangle A B C first, then triangle A D C.
When given one side and a diagonal, say B C equals 4.5 centimetres and diagonal A C equals 6.0 centimetres, you again construct right-angled triangle A B C, then another right-angled triangle A D C. The right angle is your key tool here.
The rhombus has a particularly elegant construction. Remember, its diagonals bisect each other at right angles. Given diagonals A C equals 6.0 centimetres and B D equals 4.6 centimetres, first draw A C. Construct its perpendicular bisector to find midpoint O. From O, mark points B and D along this perpendicular such that O B and O D each equal half of B D, which is 2.3 centimetres. Join all vertices, and you have a perfect rhombus.
The square follows the same method as the rhombus, since its diagonals are equal and also bisect each other at right angles. The only difference is that both diagonals have the same length.
Finally, the regular hexagon, with three beautiful methods.
Method one uses the fact that each interior angle equals 120 degrees. Start with side A B equal to 3.0 centimetres. At A and B, construct 120-degree angles. Cut off sides of 3.0 centimetres along these rays. Continue this process, always making 120-degree angles and sides of 3.0 centimetres, until you return to complete the figure.
Method two is remarkable. The side of a regular hexagon equals the radius of its circumcircle. Draw a circle of radius 3.0 centimetres. Pick any point A on the circumference. With A as centre and radius 3.0 centimetres, draw arcs cutting the circle at B and F. Repeat this process from B and F to find C and E, then from C or E to find D. You have divided the circumference into six equal parts. Join consecutive points to form your hexagon.
Method three uses central angles. Each side subtends 60 degrees at the centre, since 360 divided by 6 equals 60. Construct isosceles triangle A O B with base A B equal to 3 centimetres and angle A O B equal to 60 degrees. With O as centre and O A as radius, draw a circle. Then step around the circle using radius A B to locate points C, D, E, and F.
Let us recap the key ideas.
First, always sketch first, then construct. Second, quadrilaterals are often built by constructing triangles. Third, parallelograms exploit the properties that opposite sides are equal and diagonals bisect each other. Fourth, trapeziums use the clever trick of cutting off a segment equal to the shorter parallel side. Fifth, rectangles and rhombuses use their special angle and diagonal properties. And sixth, regular hexagons connect beautifully to circles through their 60-degree central angles and equal side-to-radius relationship.
Geometric construction is like learning a language. At first, each step feels deliberate and careful. But with practice, you will see patterns, anticipate results, and appreciate the elegant logic that connects these shapes. Keep your compass steady, measure precisely, and trust the process.
Until next time, happy constructing.