Hello, and welcome to your mathematics lesson today. We are going to explore Chapter 22: Constructions. This is where geometry meets precision — where we use only a ruler and a pair of compasses to create perfect angles, lines, and triangles. By the end of this lesson, you will know how to construct angles of various measures, bisect lines and angles, draw perpendiculars and parallels, and build triangles of different types.
Let us begin with the construction of angles.
First, how do we copy an angle? Imagine you have angle ABC, and you need to create another angle exactly equal to it somewhere else. Here is how we proceed. With B as centre, draw an arc that cuts both arms of the angle — let us call these points D and E. Now draw a new line segment PQ. With the same radius as before, draw an arc from P that meets PQ at R. Measure the distance DE with your compass, and from R, draw an arc that cuts the previous arc at S. Join P to S and extend it to T. The angle TPQ is now exactly equal to angle ABC.
Next, bisecting an angle — that means cutting it into two equal halves. Take angle ABC again. From B, draw an arc cutting AB at D and BC at E. Now, from D and E, draw two arcs with the same radius — this radius must be more than half of DE — so that they intersect at F. Join B to F. The line BF is your angle bisector, and angle ABF equals angle FBC, each being half of angle ABC.
Now let us construct some standard angles.
For 60 degrees: draw line segment OA. With O as centre, draw an arc cutting OA at B. With B as centre and the same radius, draw another arc cutting the first arc at C. Join OC and extend it to D. Angle DOA is exactly 60 degrees. This works because triangle OBC is equilateral.
For 90 degrees: start again with line OA. With O as centre, draw an arc cutting OA at B. With B as centre, same radius, cut the first arc at C. With C as centre, same radius again, cut the first arc at D. Now with C and D as centres, draw two arcs of equal radii that intersect at E. Join OE. Angle AOE is 90 degrees.
For 45 degrees, simply bisect your 90 degree angle. For 120 degrees: draw OA, then with O as centre, draw an arc cutting at C. With C as centre, draw an arc at D. With D as centre, draw another arc at E. Join OE and extend to B. Angle AOB is 120 degrees.
For 135 degrees, construct 90 degrees, then add 45 degrees by bisecting the adjacent 90 degree angle. For 75 degrees, construct 60 degrees and 90 degrees sharing the same arm, then bisect the 30 degree gap between them. With combinations like these, you can construct 105 degrees, 165 degrees, and many others.
Moving on to special constructions with lines.
To bisect a line segment AB: with A and B as centres, and radius greater than half of AB, draw arcs on both sides of AB. Let these arcs meet at C and D. Join CD. This line CD is the perpendicular bisector of AB. It cuts AB at P such that AP equals PB, and angle APC is 90 degrees. Every line segment has exactly one perpendicular bisector.
To draw a perpendicular to a line at a point P on it: simply construct a 90 degree angle at P.
To draw a perpendicular from a point P outside a line AB: with P as centre, draw an arc cutting AB at C and D. With C and D as centres, draw intersecting arcs below AB at E. Join PE, cutting AB at Q. PQ is perpendicular to AB.
To construct a line through point P parallel to line AB: mark any point Q on AB. Join PQ. At P, copy angle PQB to create angle QPR equal to it. Line through P and R is parallel to AB. This uses the fact that equal alternate angles mean parallel lines.
Now we turn to constructing triangles.
When three sides are given, say AB equals 3 centimetres, BC equals 4 centimetres, and CA equals 3.5 centimetres. First draw AB. With A as centre, radius 3.5 centimetres, draw an arc. With B as centre, radius 4 centimetres, draw another arc. Where they meet is point C. Join AC and BC. Remember: a triangle can only be constructed if the sum of any two sides is greater than the third side.
When two sides and the included angle are given: say BC equals 5 centimetres, AB equals 4 centimetres, and angle ABC equals 60 degrees. Draw BC. At B, construct angle 60 degrees. Along this new arm, mark A at 4 centimetres from B. Join AC.
When two angles and the included side are given: say AB equals 6 centimetres, angle A equals 60 degrees, and angle B equals 45 degrees. Draw AB. At A, construct 60 degrees. At B, construct 45 degrees. Where the two arms meet is C.
Let us construct isosceles triangles.
When base AB equals 5 centimetres and each base angle equals 45 degrees: draw AB, then at A and B construct 45 degrees each. Where they meet is C.
When the equal sides AB and AC equal 4 centimetres and vertex angle A equals 60 degrees: draw AB, construct 60 degrees at A, then mark C at 4 centimetres along this arm. Join BC. Interestingly, when the vertex angle is 60 degrees in an isosceles triangle, it becomes equilateral.
For an equilateral triangle with side 5 centimetres: draw AB equals 5 centimetres. With A as centre, radius 5 centimetres, draw an arc. With B as centre, radius 5 centimetres, draw another arc. They intersect at C. Join AC and BC. All three sides are equal, and all three angles are 60 degrees.
Now for right-angled triangles.
When the two sides containing the right angle are given: AB equals 4.5 centimetres, AC equals 3.5 centimetres, with right angle at A. Draw AB, construct 90 degrees at A, mark C at 3.5 centimetres, then join BC.
When one side and the hypotenuse are given: AB equals 4 centimetres, BC equals 5 centimetres, right angle at A. Draw AB, construct 90 degrees at A, then with B as centre and radius 5 centimetres, draw an arc cutting the perpendicular at C. Join BC.
Finally, let us explore circles associated with triangles.
The circumcircle passes through all three vertices of a triangle. To construct it: first draw your triangle ABC. Draw the perpendicular bisectors of any two sides. They meet at O, the circumcentre. With O as centre and OA as radius, draw the circle. It will pass through A, B, and C. OA equals OB equals OC, all being the circumradius.
The incircle touches all three sides of the triangle from inside. To construct it: draw triangle ABC. Bisect any two angles. These bisectors meet at I, the incentre. From I, drop a perpendicular to any side, say BC at P. With I as centre and IP as radius, draw the circle. It will touch all three sides.
Let us recap the key takeaways from this chapter.
First, you can construct angles of 60, 90, 45, 120, 135, and 75 degrees using ruler and compasses, and copy or bisect any given angle.
Second, you can construct perpendicular bisectors of line segments, perpendiculars from points on or outside lines, and parallels through given points.
Third, triangles can be constructed when given three sides, two sides and the included angle, or two angles and the included side — provided the triangle inequality is satisfied.
Fourth, isosceles and equilateral triangles have special construction methods based on their symmetry.
Fifth, right-angled triangles require either the two legs or one leg and the hypotenuse.
Sixth, every triangle has a unique circumcircle through its vertices and a unique incircle touching its sides, with constructions based on perpendicular bisectors and angle bisectors respectively.
Remember, construction is about precision — equal radii, careful arcs, and logical steps. Keep your compasses firm, your pencil sharp, and your mind focused on the geometric principles behind each step. Practice these constructions until they become second nature.
Thank you for listening, and happy constructing!