Hello, and welcome to today's mathematics lesson! Today, we are going to explore the fascinating world of geometric constructions using only two tools — a ruler and a pair of compasses. No protractors, no set squares, just pure geometry with these classical instruments.
By the end of this lesson, you will understand how to copy angles, bisect them, construct special angles like 60°, 30°, 90°, and 45°, work with perpendiculars and parallel lines, and construct various quadrilaterals including parallelograms, rectangles, rhombuses, and squares. Let us begin!
First, let us understand what we mean by construction. When we construct in geometry, we create accurate figures using only an unmarked ruler for drawing straight lines, and compasses for drawing circles and arcs. The beauty of this method is that every step is precise and can be justified using geometric reasoning.
Our first skill is constructing an angle equal to a given angle. Imagine you have angle ∠ABC and you need to create another angle, say ∠DEF, exactly the same size somewhere else.
Here is how we do it. First, draw a line segment EF of any convenient length. With B as centre, draw an arc that cuts both arms of the original angle. Let us say it cuts at points P and Q. Now, with E as centre and using exactly the same radius, draw an arc cutting EF at point R. Measure the distance PQ with your compasses, then with R as centre and radius equal to PQ, draw another arc cutting the previous arc at point S. Finally, join E to S and extend to point D. The angle ∠DEF you have created is exactly equal to the original angle ∠ABC. This works because we have essentially copied the triangle formed by the arc and chord!
Next, let us learn to bisect an angle — that means dividing it into two equal parts.
Given angle ∠ABC, with B as centre, draw an arc cutting both arms at D and E. Now, using D and E as centres, and with equal radii greater than half of DE, draw two arcs that intersect at point O. Join B to O and extend to P. The line BP is your angle bisector.
Notice that we require the radii to be more than half of DE. This ensures the arcs actually intersect. The result is that ∠ABP equals ∠PBC, each being exactly half of the original angle.
Now for some special angles that we can construct directly.
To construct 60 degrees, start with line segment BC. With B as centre, draw an arc cutting BC at D. Keeping the same radius, with D as centre, draw another arc cutting the first arc at E. Join B to E and extend to A. Angle ∠ABC equals 60°. This works because triangle BDE is equilateral. All sides equal the radius, so all angles are 60 degrees!
For 30°, simply bisect your 60° angle. The bisector divides it into two equal parts of 30° each.
For 90°, we extend the 60° construction. After creating point E as before, use E as centre with the same radius to create point F on the first arc. Now DE and EF are both equal to the radius, so DF spans 120 degrees. Bisect this arc by drawing intersecting arcs from E and F with radii greater than half of EF, meeting at point P. The line BP creates a 90° angle with BC. And of course, 45° is just half of 90°. Bisect your right angle to obtain it.
Moving on to line segments — let us construct a bisector of a line segment.
Given segment BC, we can find its midpoint using two methods. The first method uses congruent triangles. Construct equal angles at B and C, mark equal lengths on each arm, and join these points. Where this line crosses BC is the midpoint.
The alternative method is more direct: with B and C as centres, and radius greater than half of BC, draw arcs on both sides of the line. These intersect at points P and Q. The line PQ cuts BC at its midpoint O, with OB equal to OC, each being half of BC.
When we construct PQ this way, it is actually perpendicular to BC as well. This makes it the perpendicular bisector. So we get two properties. It bisects the segment and forms 90° angles with it.
Now let us construct perpendiculars to lines.
When the point lies on the line: with P as centre, draw an arc cutting the line at C and D. From C and D, draw intersecting arcs with equal radii greater than half of CD, meeting at O. Line PO is perpendicular to the original line.
When the point is outside the line: again draw an arc from P cutting the line at C and D. From C and D, draw arcs on the opposite side of the line from P, with equal radii greater than half of CD, meeting at Q. Join P to Q, meeting the line at O. The segment OP is your perpendicular from the external point.
Parallel lines are equally important.
To draw a line through point P parallel to given line AB, we use the property of alternate angles. Join P to any point Q on AB. At P, construct an angle equal to angle ∠PQB. This creates equal alternate angles, which guarantees parallelism. Extend this line and you have your parallel line.
Alternatively, use corresponding angles. Extend QP to R, then construct angle ∠RPD equal to angle ∠PQB. These equal corresponding angles again ensure the lines are parallel.
To construct a parallel line at a specific distance, first draw a perpendicular from any point on the original line. Mark off the required distance along this perpendicular. Then construct a perpendicular at that new point. This new perpendicular is parallel to your original line.
Now we move to constructing quadrilaterals, and we will explore several cases depending on what information is given.
When four sides and one angle are given, we start with one side. Construct the given angle at one endpoint, mark the adjacent side along this angle, then use intersecting arcs to locate the fourth vertex. For example, with sides AB, BC, CD, DA and angle A given, we draw AB, construct the angle at A, mark AD, then from D and B draw arcs with radii CD and BC respectively to find C.
When three consecutive sides and two included angles are given, we start with the middle side. Construct the two angles at its endpoints. Then mark off the remaining sides along these angle lines.
When four sides and one diagonal are given, we construct two triangles sharing that diagonal. First draw one side. Then use the diagonal and adjacent side to locate one vertex. Finally, use the diagonal and remaining sides to locate the final vertex.
When three sides and two diagonals are given, we again use triangle construction — the two triangles formed by one diagonal give us all the information we need.
When two adjacent sides and three angles are given, we start with one side. Construct angles at both ends, mark the second side, then use the third angle to find where the final side meets the first angle line.
Parallelograms have special properties we must remember: opposite sides are equal, and diagonals bisect each other.
Given two consecutive sides and the included angle, we use the fact that opposite sides equal the given sides. Construct the angle, mark both sides, then from their free ends draw arcs with radii equal to the opposite sides to find the fourth vertex.
Given two consecutive sides and a diagonal, we construct triangle ABC first using the three lengths, then locate D using the property that opposite sides are equal.
Given both diagonals and the angle between them, we use the crucial property that diagonals bisect each other. Draw one diagonal and find its midpoint. Construct a line through this midpoint at the given angle. Mark half of each diagonal along this line in opposite directions. Then join the four points.
Rectangles add the property that all angles are 90° and diagonals are equal.
Given two adjacent sides, construct a right angle at one end of the first side, mark the second side, then use intersecting arcs with radii equal to the opposite sides to complete the rectangle.
Given one side and a diagonal, construct a right angle at one end of the given side. Then use the diagonal as hypotenuse to locate the third vertex, and complete using opposite side equality.
Given both diagonals and the angle between them, remember diagonals are equal and bisect each other. Draw one diagonal and find its midpoint. Construct a line at the given angle through this point. Mark half the diagonal length in both directions. Since diagonals are equal, each half-diagonal segment is the same length.
Rhombuses have all sides equal and diagonals that bisect each other at 90°.
Given one side and one angle, draw the side, construct the angle, mark the adjacent equal side. Then from both free ends draw arcs with radius equal to the side to find the fourth vertex.
Given one side and one diagonal, the diagonal forms two triangles with the sides. Use triangle construction to find one vertex, then repeat to find the opposite vertex.
Given both diagonals, draw one, construct its perpendicular bisector, mark half of the second diagonal in both directions along this perpendicular, and join the four points. The diagonals automatically bisect each other at 90°, giving us a rhombus.
Squares combine all these properties. They have equal sides, equal diagonals, all angles 90°, and diagonals bisecting each other at 90°.
Given one side, construct a right angle, mark the equal adjacent side, then from both free ends draw arcs with the same radius to find the fourth vertex.
Given a diagonal, draw it, construct its perpendicular bisector, and mark half the diagonal length in both directions. Since diagonals are equal in a square, this locates all four vertices.
Let us recap the key takeaways from today's lesson.
First, we can copy any angle using arcs and chord lengths. We can also bisect any angle using intersecting arcs from points equidistant from the vertex.
Second, special angles 60°, 30°, 90°, and 45° can be constructed using equilateral triangles and bisection, without any protractor.
Third, perpendicular bisectors of segments give us both the midpoint and 90° angles, constructed using intersecting arcs from both endpoints.
Fourth, parallel lines are constructed using equal alternate angles or equal corresponding angles.
Fifth, quadrilaterals are built using triangle constructions. We exploit properties like opposite sides equal, diagonals bisecting each other, and specific angle measures.
Sixth, for parallelograms, rectangles, rhombuses, and squares, we always use their defining properties — opposite sides, diagonal behaviour, and angle conditions — to determine our construction steps.
Remember, every construction step can be justified with a geometric reason. The more you practice, the more these justifications will become clear, and the more confident you will become in your geometric reasoning.
Keep your compasses at the same radius when required. Ensure arcs intersect by using sufficient radius. And always check your final figure against the given conditions.
Thank you for joining this lesson on geometric constructions. Practice these methods, understand the reasoning behind each step, and you will master the art of classical geometric construction. Until next time, keep exploring the beautiful world of mathematics!