Chapter 17 — Geometrical Constructions

Introduction to instruments: Geometrical constructions are done using a small set of instruments: a compass, a ruler (straightedge) and a pencil. The compass draws arcs and circles and transfers distances from one place to another. The ruler is used as a straightedge to draw straight lines or extend segments; in constructions we do not use the ruler’s scale for measurement unless a question allows measurement. A sharp pencil produces precise marks that help produce neat intersections and accurate joins.

How to hold and use them: Hold the compass lightly but firmly at the hinge; do not press too hard so it digs into the paper. Place the needle point exactly on the point you want to use as centre and swing the pencil with a steady wrist to draw smooth arcs. When using the ruler, place it so the edge touches the two points you want to join, and draw the line with a slow steady motion. Do not use the ruler as a measuring device for constructing equal lengths except when explicitly required; instead use the compass to transfer distances.

Care of tools and workspace: Keep the compass spike sharp and the pencil lead fine so arcs meet clearly. A blunt compass makes broad marks and reduces accuracy. Tape down your paper lightly if it moves while drawing. Work on a flat, stable surface and keep your hand behind the pencil line so you can see the mark as you draw. Clean away eraser dust regularly so your arcs and intersection points remain visible.

Practice skills: Before attempting constructions, practice drawing arcs of different radii from a fixed centre and practice joining points with the ruler. Learn to set the compass to a distance between two points and then move it without changing the width. Label all construction points with capital letters and number the steps when you write the construction. These habits lead to clear, repeatable constructions and make checking much easier.

Purpose: Often you will be given a line segment AB and asked to reproduce an equal segment elsewhere without reading a measurement from a scale. This uses the compass to transfer the length directly so numerical measuring errors are avoided. The method is simple and reliable once you practice keeping the compass width unchanged.

Step-by-step method: First observe the given segment AB. Place the compass point on A and the pencil on B and open the compass to exactly the distance AB. Do not change this width. Choose a new starting point C where you want the copied segment to begin. Place the compass point at C and draw a small arc that marks a point D where the arc crosses the intended direction. The point D is the second endpoint of the copied segment. Finally, use the ruler to join C and D to form the new segment CD. By construction CD = AB because both were made with the same compass opening.

Why it works: A compass fixed at a certain width preserves that distance when moved to another location. Because the compass width was precisely the length AB, every arc drawn from C with that width locates a point at the same distance from C as B is from A. This is a basic property used across many constructions such as building triangles with given sides or copying edges of shapes.

Practical tips: Ensure you do not alter the compass setting while moving it from AB to point C. Place the compass needle exactly at the point C for correct placement. If the arc is faint, make a slightly deeper stroke but avoid pushing the compass so hard that the paper tears. Use light construction lines first and darken the final segment after verifying equality by re-measuring with the compass if needed.

Finding the midpoint — idea and use: The midpoint of a segment AB is the point M that divides AB into two equal parts. The perpendicular bisector is the line that not only divides AB into two equal parts but also makes a right angle with AB. These constructions are vital: midpoints are used to find centres of circles through two points, and perpendicular bisectors help in triangle constructions and symmetry tasks.

Construction steps for midpoint and perpendicular bisector: Choose a radius for the compass that is more than half the length of AB. With the compass centred at A draw two arcs — one above and one below the line AB. Without changing the radius, draw two arcs with centre B that intersect the previous arcs at two points; label those intersections P and Q. Now join P and Q with the ruler. The line PQ is the perpendicular bisector of AB. It meets AB at M. This point M is the midpoint, so AM = MB, and PQ is perpendicular to AB.

Why this works: The arcs from A and B with equal radius produce points P and Q that are each equidistant from A and B. Every point on the straight line PQ is equidistant from A and B; therefore the locus PQ must cross AB at the point equidistant from A and B — the midpoint M. Also, since triangles formed on either side are congruent by SSS (equal radii and equal AB halves), PQ meets AB at a right angle.

Practical notes and checks: Use a radius more than half AB so arcs intersect clearly; if arcs fail to meet, increase the radius. Keep arcs light until the joining step, then darken PQ and mark M clearly. To check accuracy, place the compass point at M and draw arcs to A and B; if AM = BM the arcs should reach both endpoints equally. Practice this construction on segments of different lengths to gain confidence.

Goal and situation: Often you will need a line through a point P on a given line l such that the new line is at right angle to l. This is the method to construct a perpendicular at a point that already lies on the line. The construction uses equal arcs and the property of perpendicular bisectors.

Detailed steps: Place the compass point at P and choose a convenient radius. Draw arcs that cut the line l at two points A and B, one on each side of P. Make sure the arc crosses l at clear and distinct points A and B. Now, with centres at A and B and any radius greater than half the distance AB, draw two arcs — one above l and one below l — so that they intersect at two points X and Y. Join X and Y with the ruler. The line XY passes through P and is perpendicular to l. Where XY meets l is the foot of the perpendicular, which in this case is exactly P because P lies between A and B and we used P to draw the initial arc.

Reasoning: The points X and Y are found so that XA = XB and YA = YB (they lie on perpendicular bisector of AB). The perpendicular bisector of AB passes through points equidistant from A and B. Because P was chosen so that A and B are symmetric around it on l, the perpendicular bisector passes through P and meets l at right angle. Thus XY is the required perpendicular.

Accuracy hints: Use a radius for the second pair of arcs that ensures clear intersection points X and Y. If arcs do not intersect, increase the radius. Keep the compass width steady. Draw construction arcs lightly and darken the final perpendicular only after checking that it passes through P and appears to make a right angle. A small square mark may be added at the corner to indicate the right angle for presentation.

When we need a perpendicular from an external point: Suppose you have a line l and a point P not on l. The task is to drop the shortest path from P to l, which is the perpendicular. This construction finds the foot of the perpendicular using circles and perpendicular bisectors.

Stepwise construction: Place the compass point on P and draw an arc that cuts line l at two points A and B. These are the two points on l such that PA = PB; choose the radius so that both intersections A and B are clear. Now set the compass to a radius greater than half AB. With centres at A and B draw arcs that intersect on the side of the line closer to P at point Q. Join P and Q with the ruler. The line PQ will meet l at H; PQ is perpendicular to l and H is the foot of the perpendicular from P to l.

Why it works: The intersection Q lies on the perpendicular bisector of AB, hence Q is equidistant from A and B. The line joining P and Q joins a point P (with PA = PB) to a point Q on the perpendicular bisector and therefore must meet l at right angle; the construction enforces the shortest distance from P to l by symmetry and congruence of triangles formed in the process.

Checks and tips: If two intersections of arcs from A and B appear (one on each side of l), choose the one on the same side as P so PQ meets l between A and B. Make the initial arc from P large enough to guarantee two intersections. After drawing PQ, verify by placing the compass at H and checking equal distances to A and B, or by using a protractor if allowed. Keep marks light until final verification, then darken the correct perpendicular line.

Why 60° is basic: The 60° angle is fundamental because it appears at each vertex of an equilateral triangle. Using a compass and ruler we can construct a 60° angle without a protractor. The same idea also builds an equilateral triangle given one side. This construction rests on using equal radii from two points to locate a third point equidistant from both.

Constructing a 60° angle at point O: Draw a ray OA that will be one side of the required angle. Choose any radius and with center O draw an arc that cuts OA at point B. Keeping the same radius, place the compass point at B and draw another arc that intersects the first arc at point C (inside the plane region where you need your angle). Join O to C using the ruler. The angle AOC is 60°. This works because triangle OBC formed by equal radii OB = BC = OC is equilateral in effect; the construction creates equal distances and hence equal angles at O.

Constructing an equilateral triangle given AB: Given segment AB as one side, open the compass to length AB and draw an arc with center A, and another arc with center B using the same radius. The two arcs meet at two possible points; choose one of them as C. Join A to C and B to C. Triangle ABC is equilateral since AC = AB = BC by construction. Each interior angle of ABC is therefore 60°.

Practical tips and checks: Use the same compass width when you draw the second arc from B as you used from O; do not change the setting. If the arcs intersect in two places, either intersection gives a valid 60° angle or equilateral triangle (one above and one below the base). After drawing triangle ABC, you can check by measuring sides with the compass to confirm equality. Practice this construction with different base lengths to become confident in setting and transferring compass widths accurately.

Purpose: Copying an angle means reproducing a given angle at another location and orientation while keeping the opening exactly the same. This is used when making similar shapes or when we need to construct parallel lines by copying corresponding angles.

Method (arc-and-chord): Let the given angle be ∠XOY. Place the compass at O and draw an arc that cuts both sides of the angle at points A and B. This arc records a pair of points on the rays whose separation depends on the angle. Without changing the compass width, transfer the compass to the new point P where you want the copied angle. Draw a similar arc centered at P to intersect the chosen initial ray PR at S. Now measure the distance AB on the first arc with the compass (open the compass to exactly the distance between A and B). With the compass set to this chord length, place the point at S and draw an arc which intersects the arc drawn from P at T. Join P to T. Then ∠RPT is equal to ∠XOY because the chord AB determines the same opening when reproduced on the new arc.

Why it works: The first arc forms two marked points A and B which lie on the sides of the angle at equal radius from the vertex; the distance AB depends only on the angle and the radius. Reproducing the same arc radius and chord distance at the new location ensures the rays enclose the same angle. This method avoids protractors and relies only on basic compass properties.

Tips: Use clear arcs and keep the compass width fixed while transferring AB. Make small light arcs first and then darken the final rays. Practice copying both small and large angles to gain confidence. If the chord AB is short, increase the initial arc radius to make AB easier to transfer accurately.

Objective and use: Bisecting an angle means constructing a ray that divides the angle into two equal parts. Angle bisectors are used in triangle constructions and several geometry problems; for example, the in-centre of a triangle is the intersection of its three angle bisectors.

Step-by-step construction: Given angle ∠AOB, place the compass at O and draw an arc that cuts both rays OA and OB at points C and D respectively. The exact radius is not important but should be large enough so C and D are distinct and clearly visible. Now, with the compass set to a radius more than half CD, draw arcs with centers at C and at D such that the two arcs intersect at a point E inside the angle. Join O and E with the ruler. The ray OE is the bisector of angle AOB and divides it into two equal angles ∠COE and ∠EOD.

Reasoning behind the steps: Points C and D are at the same distance from O by construction. The intersection point E of arcs centred at C and D is equidistant from C and D. Therefore line OE joins the vertex O to a point E that is equidistant from the two arms of the angle, which is the definition of an angle bisector. The congruence of triangles formed ensures ∠COE = ∠EOD.

Checks and hints: Choose an initial arc radius that gives C and D well separated; otherwise the second arcs may not intersect. Keep the compass width unchanged while drawing the second pair of arcs. After constructing OE, you can verify by drawing a small arc from O that cuts OE and the arms and checking that the chord lengths on each side are equal when measured with the compass. Label E and darken the bisector for clarity.

Understanding parallel lines: Two lines are parallel if they lie in the same plane and never meet. One practical way to construct a line parallel to a given line l through a point P (not necessarily on l) is to copy corresponding or alternate interior angles formed by a transversal.

Method by copying corresponding angles: Choose any point A on the given line l and join A to the external point P by drawing the segment AP. Now at A construct the angle that AP makes with l — this is possible by drawing a small arc centered at A that crosses both l and AP, marking the chord points. Transfer this arc and chord to point P as in the angle copying method: draw a similar arc from P and reproduce the chord length. The ray at P that corresponds to the ray on A will form an angle equal to the angle at A. Because corresponding angles formed by the transversal AP are equal, the new line through P is parallel to l. Join the point P with the new ray point to draw the parallel line through P.

Alternate method using perpendiculars: Another method is to draw a perpendicular to l at some point and then draw a perpendicular to this perpendicular through P; the second line will be parallel to l. For example, at a point A on l construct a perpendicular m. Then through P construct a perpendicular to m. This new line is parallel to l because both are perpendicular to the same line m.

Checks and tips: Whichever method you use, verify parallelism by measuring corresponding angles with a protractor if allowed, or by drawing a transversal and checking that alternate interior angles are equal. Keep the compass width steady and use light arcs until the final line is confirmed; then darken the parallel line. Practise both methods so you can choose the simpler one in different problems.

Triangles from given sides: In class 6 we practice constructing triangles where sides are given in a simple way, such as an equilateral triangle (all sides equal) or an isosceles triangle (two sides equal). These constructions use the compass to copy lengths and locate the third vertex by intersecting arcs.

Constructing an equilateral triangle: Start with the given side AB. Set the compass to the length AB. With centre at A draw an arc, and with the same radius and centre at B draw another arc. The two arcs meet at point C (there are two possible intersections; choose one). Join A to C and B to C. Triangle ABC is equilateral because AC = AB = BC by construction. Each interior angle will be 60° automatically.

Constructing an isosceles triangle with a given base and equal side length: Given base BC and a required side length l for the equal sides, draw BC first. With centres at B and C and radius l draw arcs that intersect at A. Join A to B and A to C. Then AB = AC = l, making triangle ABC isosceles with base BC. Base angles at B and C are equal for an isosceles triangle.

Constructing from base and vertex angle (basic approach): If a base BC and the vertex angle at A are given, first draw base BC, then copy the given angle at one endpoint using the angle-copying method; repeat at the other endpoint if needed to find the intersection point A. For class 6, practice often uses equal-side constructions where using the compass for distances is straightforward.

Verification and neatness: After constructing, use the compass to check side equalities. Mark equal sides with small dashes for clarity. Draw light construction arcs first and darken the final triangle after verification. Label the vertices clearly as A, B and C.

Quick recap of copying a segment: Copying a line segment is one of the most frequently used operations in constructions. It is done by opening the compass to the length of the given segment and drawing an arc from the new start point to locate the matching endpoint. This transfers the exact length without numeric measurement.

Procedure and uses: Given segment AB and a new starting point P, place the compass point at A and open it to B. Without changing the width, move the compass to P and draw an arc cutting the intended direction at Q. Join P and Q to form PQ equal to AB. This method is used as a building block in constructing triangles (when one or more sides are given) and in drawing polygons with equal sides. It is also useful in practical situations such as designing simple patterns, copying pieces in craft work, creating scale diagrams and marking distances on maps where uniform lengths are required.

Combining with other constructions: Copied segments often serve as sides while you construct angles, perpendiculars, or parallel lines to finish the figure. For example, to draw a rectangle with one side known, copy the side length at the opposite location to get the parallel opposite side. To make a rhombus of side a, copy the side length successively around the shape and use arc intersections to locate vertices symmetrically.

Practical hints and presentation: Keep the compass width unchanged while transferring. Work on a stable, taped sheet so the paper does not slip. Use light construction arcs and then darken the final required lines. Label all points and write short construction steps so someone else can reproduce your figure. Regular practice will improve speed and neatness of copying segments accurately.

From simple steps to complex shapes: Most complex constructions are just a sequence of the basic operations already learned: copying segments, drawing perpendiculars, bisecting angles, and drawing parallel lines. Knowing the small steps and the reasons they work helps you plan multi-step constructions like rhombuses, rectangles, kites and some special triangles.

Example — constructing a rhombus of side a: Start by drawing one side AB of length a. With centres A and B and radius a draw arcs that intersect at C (choose one intersection). Join A to C and B to C to create triangle ABC with AB = BC = a. Now use the same radius a with centre C and centre B (or use another systematic step) to locate the fourth vertex D such that CD = DA = a and AB = BC = CD = DA. Joining the points in order yields rhombus ABCD. If required, draw diagonals and note that they bisect each other at right angles, which you can verify by constructing perpendicular bisectors. Each step uses a basic construction repeated carefully.

Example — constructing a rectangle: Draw one side AB. At A and B draw perpendiculars to AB. On one perpendicular mark a point C such that AC equals the required width, measured by copying the length with the compass. From that point copy the same length to the other perpendicular and join to form the opposite side. This method combines perpendicular construction and copying segments to get a rectangle with right angles and equal opposite sides.

Strategy and checking: Before starting the paper work, write the plan in steps so you do not lose track. Draw light construction arcs and label every intermediate point. After constructing, check lengths with the compass and verify right angles or parallelism by the constructions used. If an error is found, erase only the immediate incorrect part and reconstruct that step rather than starting over. Practise combining different methods so you can choose the easiest path in varied problems.

Importance of accuracy: Geometry constructions must be neat and accurate so that the relationships shown are true and can be checked. Small errors in compass width, shaky arcs or incorrect joining points will change the figure and lead to wrong conclusions. Students must develop careful habits to avoid mistakes and to present constructions clearly in exams.

Work habits for accuracy: Use a sharp pencil and a well-adjusted compass. Keep the compass hinge tight enough so the width does not change while moving it, but not so tight that it is hard to swing. Place the paper on a firm flat surface and fix it with tape if needed to prevent slipping. Draw arcs lightly first; these are construction guides. When joining points with the ruler, hold the ruler steady and draw the line in one continuous motion to avoid wobbles.

Presentation of construction steps: Write the construction steps in clear numbered sequence and label all points with capital letters. Show construction arcs and intermediate points when the question asks for construction; do not erase them. Darken only the final required lines or shapes for clarity. If asked to state the reason for each step, write short notes such as "PQ is perpendicular bisector so AM = MB".

Common mistakes and how to avoid them: A frequent error is changing the compass width during transfer — verify by rechecking the original segment width before drawing. Another error is choosing a compass radius too small so arcs do not intersect; if arcs fail to meet, increase the radius. Using the ruler’s scale to measure when a pure ruler-and-compass method is required is also incorrect. Always label points to avoid confusion when joining them. If you make a small mistake, erase carefully and redo only the affected part.

Checking constructions: After completing the figure, verify equalities by using the compass to compare lengths, check right angles by constructing a small perpendicular, or check parallel lines by drawing a transversal and measuring corresponding angles if allowed. Regular practice and careful checking will make constructions faster and more accurate over time.