Hello, and welcome to your mathematics lesson. Today, we begin Chapter 19: Constructions. In this chapter, we will learn how to create precise geometric figures using simple tools — a ruler, a compass, and set-squares. We will discover how to copy angles, bisect them, construct special angles, draw perpendiculars, and even build complete triangles from given measurements. Let us start our journey into the art of geometric construction.
Section 19.1: Using a Ruler and a Compass.
A ruler and a compass are the classical tools of geometry. With these two instruments, we can perform remarkable feats of precision. We can copy any given angle exactly. We can cut an angle into two equal halves — this is called bisection. We can construct specific angles like 60 degrees, 90 degrees, or 45 degrees, starting from any point we choose. We can find the exact middle of a line segment by drawing its perpendicular bisector. We can draw perpendicular lines — both from points lying outside a line, and from points sitting directly on the line. Each of these skills builds upon the others, so let us explore them step by step.
Section 19.2: Construction of Angles.
First, let us learn to copy a given angle. Imagine you have angle ∠AOB, and you need to create an identical angle at a new point Q. Begin by drawing a line segment QR of any convenient length starting at Q. Now, place your compass point at O, the vertex of your original angle, and draw an arc that cuts both arms of the angle at points C and D. Without changing your compass width, move to point Q and draw a similar arc that cuts QR at point T. Next, measure the distance between C and D with your compass. Transfer this distance by placing your compass point at T, and draw an arc that intersects your previous arc at point S. Finally, draw a line from Q through S and extend it to point P. The angle ∠PQR you have created equals the original angle ∠AOB. The secret lies in preserving distances — the arc radii remain constant, ensuring the angles match perfectly.
Now, let us bisect a given angle — that is, divide it into two equal parts. Take angle ∠AOB as your starting point. With O as centre, draw an arc that cuts both arms at points C and D. From C and D, draw two new arcs using the same radius — but this radius must be more than half the distance between C and D. Let these arcs meet at point E. Draw the line OE. This line bisects the angle, meaning ∠AOE = ∠BOE. The condition about the arc radius is crucial — if your arcs are too small, they will not intersect, and your construction will fail.
Next, we construct specific angles from scratch.
To construct 60 degrees: draw line segment OA. With O as centre, draw an arc cutting OA at B. Keeping the same compass width, place the point at B and draw another arc cutting the first arc at C. Join OC and extend to D. Angle ∠DOA equals 60 degrees. This works because triangle OBC is equilateral — all sides equal, all angles 60 degrees.
To get 30 degrees, simply bisect your 60-degree angle. Each half becomes 30 degrees.
For 90 degrees, I will show you the first method. Start with line segment OA. With O as centre, draw an arc cutting OA at B. From B, with the same radius, draw an arc cutting the first arc at C. From C, with the same radius again, draw another arc cutting the first arc at D. Now from C and D, draw two arcs of equal radii that intersect at E. Join OE. Angle ∠AOE equals 90 degrees. The line OE is perpendicular to OA. We write this as OE ⊥ OA.
For 45 degrees, construct 90 degrees first, then bisect it. Each portion becomes 45 degrees.
To construct 120 degrees: start with OA, draw an arc cutting it at C. From C, draw an arc of the same radius cutting the first arc at D. From D, draw yet another arc of the same radius, cutting the first arc at E. Join OE and extend to B. Angle ∠AOB equals 120 degrees. Notice that 120 equals 180 minus 60 — you are essentially marking off two 60-degree portions from the straight line.
For 135 degrees, first construct 90 degrees, then add 45 degrees by bisecting the adjacent 90-degree angle. Your final angle becomes 90 plus 45, which is 135 degrees.
For 75 degrees, combine 60 degrees and 15 degrees. Construct 90 degrees and 60 degrees at the same point. The gap between them is 30 degrees. Bisect this 30-degree gap to get 15 degrees. Add this 15 degrees to your 60 degrees, giving 75 degrees.
With these building blocks, you can construct many more angles. 105 degrees equals 90 plus 15. 150 degrees equals 90 plus 60, or 120 plus 30. The possibilities expand through creative combinations.
Section 19.3: Construction of Perpendiculars.
First, the perpendicular bisector of a line segment. Given segment AB, we want a line that cuts AB exactly in half, at right angles. With A and B as centres, draw arcs of equal radii on both sides of AB. These radii must exceed half of AB's length. Let the arcs intersect at points C and D. Join CD, which cuts AB at M. Now AM equals BM, and angle ∠AMC equals 90 degrees. Thus CD is the perpendicular bisector — it bisects AB and is perpendicular to it.
Next, drawing a perpendicular from a point outside a line. Given line AB and external point C. With C as centre, draw an arc cutting AB at P and Q. From P and Q, draw arcs of equal radii intersecting at D on the opposite side of AB from C. Join CD, cutting AB at M. CM is your required perpendicular.
Finally, a perpendicular through a point on the line itself. Given line AB with point M on it. With M as centre, draw two arcs of equal radii cutting AB at P and Q. From P and Q, draw intersecting arcs at point X. Join MX. This is your perpendicular through M.
Section 19.4: Using Set-Squares.
Set-squares are triangular tools, typically made of plastic or metal. There are two standard types. The first has angles 45 degrees, 90 degrees, and 45 degrees — an isosceles right triangle. The second has angles 30 degrees, 90 degrees, and 60 degrees.
These tools make angle construction rapid and accurate. Simply align the appropriate edge with your base line, and draw along the other edge. You can create 30, 45, 60, 75, 90, and 105 degrees directly. For 75 degrees, combine the 45-degree and 30-degree set-squares. For 105 degrees, combine 60 degrees and 45 degrees.
Set-squares also draw perpendiculars efficiently. To drop a perpendicular from an external point C to line AB: place your set-square so its 90-degree edge lies along AB, and the perpendicular edge passes through C. Draw along this edge to meet AB at N. CN is perpendicular to AB.
For a perpendicular through a point C on the line: place the 90-degree edge of your set-square along AB at point C, and draw along the perpendicular edge. The resulting CD is perpendicular to AB at C.
Section 19.5: Construction of Triangles.
We can construct a triangle when given specific combinations of measurements. There are three fundamental cases.
Case one: three sides given. Suppose we need triangle ABC with CB equal to 4 centimetres, AC equal to 6 centimetres, and AB equal to 7.6 centimetres. First, draw a rough sketch to visualise the arrangement. Draw the longest side AB equal to 7.6 centimetres. With A as centre, draw an arc of radius 6 centimetres. With B as centre, draw an arc of radius 4 centimetres, intersecting the first arc at C. Join AC and BC. Triangle ABC is complete.
But heed this warning: three lengths form a triangle only if the sum of any two sides exceeds the third side. If AB equals 7 centimetres, BC equals 4 centimetres, and AC equals 3 centimetres, no triangle exists. Here, BC plus AC equals exactly 7 centimetres, which equals AB. The points collapse onto a straight line — no triangle forms.
Case two: two sides and the included angle given. Construct triangle ABC with AB equal to 3 centimetres, BC equal to 5 centimetres, and angle ∠ABC equal to 60 degrees. Draw BC equal to 5 centimetres first. At B, construct angle PBC equal to 60 degrees. With B as centre, draw an arc of 3 centimetres cutting BP at A. Join AC. Your triangle is complete.
Case three: two angles and the included side given. Construct triangle ABC with AB equal to 4 centimetres, angle A equal to 60 degrees, and angle B equal to 30 degrees. Draw AB equal to 4 centimetres. At A, draw line AP making 60 degrees with AB. At B, draw line BQ making 30 degrees with AB. Let AP and BQ meet at C. Triangle ABC is your required triangle.
Let us now recap the key takeaways from this chapter.
First, with a ruler and compass, you can copy angles, bisect angles, construct standard angles, and create perpendicular bisectors and perpendicular lines from various positions.
Second, angles like 30, 45, 60, 90, 120, and 135 degrees build from basic constructions, often by bisection or combination of simpler angles. Third, set-squares provide quick, accurate methods for standard angles and perpendiculars, complementing compass constructions.
Fourth, triangles construct from three sides, two sides with included angle, or two angles with included side — but three sides must satisfy the triangle inequality.
Fifth, precision matters: compass widths must be maintained, radii must exceed half-distances where specified, and constructions must be executed in proper sequence.
Sixth, perpendicularity and bisection are powerful geometric concepts that appear repeatedly across different construction types.
Practice these constructions slowly and deliberately. Geometric construction rewards patience with elegance and precision. Until our next lesson, keep your compass steady and your lines true. Goodbye, and happy constructing!