Chapter 14 — Practical Geometry

Introduction to instruments: Practical geometry depends on a few simple instruments. The basic set for Class 6 includes a ruler (or scale), a compass, a protractor and a set-square. Each tool serves special tasks and must be used with care for accurate drawings. Learning the correct way to use them is the first step to good constructions.

Ruler/Scale: A ruler is used to draw straight lines and to measure lengths in centimetres and millimetres. When measuring, always align the zero mark exactly at the starting point. Hold the ruler firmly so it does not slip, and draw the line with the pencil point close to the edge. Use the millimetre marks when more precision is needed.

Compass: A compass draws circles and arcs and is used to transfer distances from one part of a figure to another. Fix the needle point at the centre point and loosen the pencil holder just enough so the lead draws a clean arc. Keep the compass upright and rotate it smoothly around the needle; avoid changing the compass width while drawing linked arcs.

Protractor: A protractor measures and helps to draw angles in degrees. Place its centre hole exactly on the vertex and line up its baseline with one arm of the angle. Read carefully on the correct scale—protractors usually have two sets of numbers (inner and outer). Choose the one that starts at the arm being used as zero.

Set-square: Set-squares are triangular tools that give standard angles such as 30°, 45° and 60° and help draw perpendiculars and parallel lines fast. Use a set-square together with a ruler for longer parallel lines by sliding it along the ruler without tilting.

Care and maintenance: Keep points sharp: a sharp pencil and a fine compass point give clear lines and accurate intersections. Avoid pressing too hard while drawing. Keep instruments in a box to protect the protractor from cracks and the set-square from warping. When using a compass, close it slightly before storing to avoid accidents. Practise on scrap paper to gain control before drawing final constructions.

Basic terms: A point marks a position and has no size. A line extends in both directions without end; it is shown with arrows at both ends. A line segment has two endpoints and a fixed length. A ray starts at one point and goes infinitely in one direction.

Drawing and naming: Use capital letters to name points, for example A, B, C. To show a line passing through A and B, write AB with a double arrow above when referring to the line. For a segment between A and B write AB with a bar above. To mark a ray starting at A and passing through B write AB with a single arrow above pointing right.

Measuring segments: Measure a segment using a ruler: align the zero with one endpoint and read at the other endpoint. Record lengths in cm or mm. For more precision use millimetres.

Constructing segments: To construct a segment of a given length, mark one endpoint, align the ruler and mark the other endpoint at the required length, then join the points. When using a compass, set it to the required length and transfer it.

Practice points: Distinguish clearly in drawings whether a figure shows a line, ray or segment. Lines and rays cannot be measured as they are infinite; measure only segments. Correct notation and neatness are important in geometry work.

What is an angle: An angle is formed by two rays with a common starting point called the vertex. We write the angle with three letters: for example ∠ABC has vertex at B with rays BA and BC.

Measuring angles with a protractor: Place the protractor so that its centre hole is exactly at the vertex. Make sure the base line of the protractor aligns with one arm of the angle. Read the scale where the other arm meets the protractor. Use the inner or outer scale correctly depending on the direction of the angle.

Types of angles: An acute angle is less than 90°, a right angle is exactly 90°, an obtuse angle is between 90° and 180°, and a straight angle is 180°. Recognize them by measurement or by shape.

Drawing angles: To draw an angle of a given measure first draw one ray. Place the protractor centre on the end of the ray and mark the degree point on the paper. Remove the protractor, join the vertex to the marked point and extend to form the second arm.

Careful steps and checking: Always place the protractor flat, line up zero correctly and make a small dot for accuracy. After drawing, measure the angle to confirm the correct degree. Practice drawing several angles to get comfortable with reading the protractor scales and handling instruments.

Meaning and objective: An angle bisector divides an angle into two equal parts. Constructing an angle bisector precisely with compass and ruler is a standard exercise that teaches how to use arcs and congruent triangles. The bisector does not require measuring degrees and gives an exact division based on symmetry.

Step-by-step construction (compass method): Place the compass point at the angle's vertex O and draw an arc that cuts both arms at points P and Q. Without changing the compass width, draw two arcs: one with centre P and one with centre Q so that these arcs intersect at a point R inside the angle. Now draw the ray OR. This ray is the bisector of the angle because it divides the original angle into two equal angles.

Geometric reason: When arcs with the same radius are drawn from P and Q, the intersection point R is equidistant from P and Q. Triangles OPR and OQR have OP = OQ (both are radii of the first arc), PR = QR (both radii of the second arcs) and OR is common; by SSS congruence the triangles are congruent. Corresponding angles at O in these triangles are equal, so OR bisects the angle at O.

Properties and extension: The bisector creates two angles each equal to half the original. It is unique inside the angle. If you draw a circle centred at O, the bisector also divides the arc between the two arms into two equal arcs. Angle bisectors are used in many constructions, for example to locate points equidistant from two lines and to find in-centres of triangles.

Accuracy tips and checks: Use a comfortable compass width so arcs clearly intersect; very small arcs can be hard to see. Mark intersection points clearly and draw straight ray through vertex and intersection. To check, measure the two angles with a protractor; both should read the same value. Practice bisecting angles of different sizes and writing the construction steps neatly when answering questions.

Perpendicular lines: Two lines are perpendicular when they meet at a right angle (90°). We show this with a small square at the intersection point in a drawing. Perpendiculars are common in shapes like rectangles and squares.

Constructing a perpendicular from a point on a line: To draw a perpendicular at point A on line l, place the compass at A and draw arcs to cut the line at two points P and Q on either side. With the same compass width, draw arcs with centres P and Q above (and below) the line so they meet at R (and S). Join A to R; AR is the perpendicular to l.

Constructing a perpendicular from a point not on the line: From external point B, draw an arc that meets line l at P and Q. With centres P and Q and equal radius draw two arcs meeting at R above the line. Join B to R; BR is perpendicular to l.

Perpendicular bisector of a segment: To construct the perpendicular bisector of segment AB, use a compass with width more than half AB. Draw arcs with centres A and B above and below the segment so the arcs intersect at two points. Join these intersection points; this line cuts AB at its midpoint M and is perpendicular to AB.

Why bisector works: The intersection points are equidistant from A and B, so the joining line has all points equidistant from A and B; its intersection gives the midpoint and creates right angles by symmetry.

Parallel lines: Two lines are parallel if they lie in the same plane and never meet, however far they are extended. Parallel lines are marked with small arrows in figures.

Drawing a parallel through a given point: Given line l and a point P not on l, we can draw a line through P parallel to l using a set-square and ruler or by copying corresponding angles with a protractor.

Using set-square (sliding method): Place one set-square along line l and the other against it to form a right angle or a known angle. Slide the pair until the second set-square passes through P while keeping the first on line l. Draw along the second set-square through P to get the parallel.

Using alternate interior angles: Choose a point A on line l and draw a transversal from A to P. Measure the angle it makes with l using a protractor. At P construct the same angle on the opposite side of the transversal. The new line through P will be parallel to l because the corresponding or alternate interior angles are equal.

Properties and checks: Parallel lines maintain equal corresponding and alternate angles with a transversal. Verify parallelity by measuring these angles or by seeing that the two lines never meet on paper in the drawing range. Mark parallel lines with small arrow symbols for clarity.

Copying a line segment: To copy a given segment AB onto a different location, use a compass. Place the compass point at A and the pencil at B and set the width. From a new point C draw an arc with the same compass opening and mark point D where the arc meets the chosen direction. Connect C and D to get segment CD equal to AB.

Steps and care: Keep the compass width unchanged while transferring. Make sure the starting point C is fixed and the pencil mark is precise. Use a ruler to draw the direction line only if needed; the compass gives the exact length.

Copying an angle: To copy an angle ∠PQR at a new vertex S, first draw a ray from S as one arm. With the compass at Q draw an arc to cut both arms of the original angle at A and B. With the same radius draw an arc from S to cut the new ray at A'. Now set the compass to the distance AB and with centre A' draw an arc to mark B'. Join S to B'. The angle ∠B'SA' is a copy of ∠PQR.

Reason and checks: The arcs preserve distances from the vertex and between the intersection points on the arms, so the constructed angle has the same opening. Check by measuring both angles with a protractor; they should match.

Practice notes: Accurate arc drawing and keeping compass width fixed are critical. Label points clearly and write steps when asked in exams to show understanding.

Triangle construction basics: A triangle is determined by its three sides or by two sides and the included angle among other combinations. The common easy constructions for Class 6 are SSS (three sides given) and SAS (two sides and included angle given).

SSS construction (three sides): Given lengths a, b, c for sides BC, CA, AB. Draw one side, say AB, of length c using a ruler. With centre A draw an arc of radius b. With centre B draw an arc of radius a. Their intersection gives point C. Join C to A and C to B to complete triangle ABC. The three lengths meet at the intersection because the distances from centres are fixed.

SAS construction (two sides and included angle): Given two sides AB and AC and included angle ∠A. At point A draw the given angle. On one arm mark AB of the given length. On the other arm mark AC of its given length. Join the end points to form triangle ABC. The included angle fixes how the sides meet.

Tips for accuracy: Use clear ruler lines for the initial side and precise compass arcs for intersections. Label every point. If arcs do not meet, re-check compass width and starting points. For obtuse or nearly flat triangles, make larger compass arcs for better intersection visibility.

Checking the triangle: Measure all sides with the ruler to confirm they match the given values. Also measure angles if needed. Writing construction steps neatly is required in exams.

ASA construction (two angles and included side): If two angles and the included side are given, we can construct the triangle uniquely. Suppose side BC and angles ∠B and ∠C are given. Draw BC of given length. At B and C construct the given angles using a protractor. The two rays from B and C will meet at A. Join A to B and A to C to complete triangle ABC.

RHS construction for right triangles: RHS stands for Right angle, Hypotenuse and Side. If a right triangle has its hypotenuse and one side given, it can be constructed uniquely. Draw the hypotenuse AB. At one end (say A) construct a right angle. On the arm of that right angle mark the given side length AC. Join C to B to complete the triangle. Alternatively use circle properties: the right angle subtends a diameter, so a circle of diameter AB will have any point on it forming a right triangle.

Care and checking: For ASA, ensure both angles are constructed inside the same side of BC so their rays meet. For RHS check the right angle with a set-square and ensure the side length is placed on the correct arm. Always label the triangle and measure to verify.

Why uniqueness holds: For ASA, knowing two angles and the included side fixes the third angle and thus the triangle shape. For RHS, a right angle and hypotenuse fix the position of the third vertex on a circle with diameter equal to the hypotenuse, and the side length determines the exact point.

Square construction: A square has four equal sides and four right angles. To construct a square of side 'a', first draw side AB of length a. At A construct a right angle using a set-square and draw ray AX. Using the compass, from A mark point D on AX such that AD = a. Now at point B construct a right angle and draw a ray BY. From B mark point C on BY such that BC = a. Finally join C to D to complete square ABCD. Check all sides equal and corners right angles.

Rectangle construction: A rectangle has opposite sides equal and all angles equal to 90°. To construct rectangle with length l and breadth b, draw side AB = l. At A draw a perpendicular and mark AD = b. At B draw a perpendicular and mark BC = b on that perpendicular. Join C to D. Verify AB = CD = l and AD = BC = b and all angles are right angles.

Using set-square and ruler: Use set-square to draw right angles accurately. Use parallel construction to ensure opposite sides are parallel: once AD is drawn, use set-square sliding to draw a line through D parallel to AB, and similarly from C to meet it. This guarantees the figure is a rectangle.

Checks and practice: Measure all sides and angles to confirm the figure is a square or rectangle. Mark right angles with small squares in the drawing. Practice with different sizes to gain speed and accuracy.

Rhombus construction: A rhombus has all four sides equal but angles may not be right. To construct a rhombus with side 'a' and one angle θ, draw side AB = a. At A construct angle θ and draw ray AX. With centre B and radius a draw an arc. With centre A and radius a draw an arc cutting AX at D. The arcs ensure AD and BC each equal a. Join D to C and C to B to complete rhombus ABCD. Check all sides equal using the ruler and mark opposite sides parallel.

Parallelogram construction: A parallelogram has opposite sides equal and parallel. Given two adjacent sides a and b and angle θ between them, draw side AB = a. At A construct angle θ and on that ray mark AD = b. Through B draw a line parallel to AD (use set-square or copy angle) and through D draw a line parallel to AB. Their intersection gives point C. Join C to B and C to D to finish ABCD which is a parallelogram.

Properties and checks: Opposite sides are equal and parallel in both rhombus and parallelogram. Diagonals bisect each other in a parallelogram; in a rhombus diagonals also are perpendicular. Verify these properties by measuring or drawing diagonals and checking midpoints.

Practical tips: Accurate angle copying and parallel construction are essential. Label each step when writing construction answers and always re-check side lengths and parallel marks.

What is a locus: A locus is a set of points that satisfy a given condition. In plane geometry, loci are usually described as points at a fixed distance from a given point (a circle), points equidistant from two points (perpendicular bisector), or points at a fixed distance from a line (pair of parallel lines).

Common loci: (1) Points at distance r from point O form a circle with centre O and radius r. (2) Points equidistant from two points A and B lie on the perpendicular bisector of AB. (3) Points equidistant from two intersecting lines lie along the angle bisectors of those lines. (4) Points at fixed distance d from a line l form two lines parallel to l at distance d on either side.

Using loci to solve problems: Many construction problems ask to find a point that satisfies two conditions at once. The solution is the intersection of two loci. For example: to find points equidistant from A and B and at distance r from C, draw the perpendicular bisector of AB and the circle centred at C with radius r. Their intersection points, if any, are solutions.

Construction steps and practice: Learn how to draw each basic locus cleanly: circle with compass, perpendicular bisector, parallel lines and angle bisectors. To solve a combined problem, draw both loci and find their intersection. If loci do not meet, state that there is no solution. Label intersection points and describe why they satisfy both conditions.

Checks and reasoning: After locating a solution point, verify distances with ruler or compass. Explain in words why the intersection point meets both requirements: cite the definition of each locus used.

Why checking matters: A construction is only useful if it is accurate. Small slips in setting the compass or aligning the ruler can produce errors. Checking helps find mistakes and teaches students how to reason about constructions.

Methods of checking: After construction, measure the required lengths with a ruler to ensure they match. Use a protractor to measure angles when appropriate. For perpendiculars and parallels, use a set-square to check right angles and parallelism. For triangle constructions, measure all three sides to confirm SSS or the given parts like SAS or ASA.

Correcting common errors: If arcs do not meet in SSS construction, re-check the compass widths and the starting positions. If an angle copy is off by a few degrees, repeat the arc steps and ensure the compass width is unchanged. If two lines meant to be parallel meet, check that the corresponding angle was copied on the correct side of the transversal.

Recording steps and justification: In tests, always write the construction steps in order and mention checks performed. State why an intersection point solves the problem — for example, explain why it is equidistant or why it bisects an angle. This shows understanding beyond just drawing.

Practice habit: Get into the habit of lightly drawing construction lines and darkening the final figure. Use an eraser to remove unnecessary arcs after checking. Neat labels and clear ticks/arrows help in checking and in exam presentation.

Importance of presentation: In geometry, a clear drawing and correct labels make the construction easy to read and check. Examiners look for both correct construction steps and a tidy, well-labelled figure. Good presentation reduces the chance of mistakes and helps communicate your method clearly.

Labelling rules and conventions: Use capital letters for points in the order you construct them (A, B, C...). Label the centre of circles as O, and mark intersection points clearly with capital letters. Use small tick marks on sides to show equality, use a small square symbol to denote right angles and use arrow marks to indicate parallel sides. Write the name of the figure (for example triangle ABC) near the drawing so the reader immediately recognises it.

Line darkness and erasing: Draw construction lines and arcs lightly. These helper lines show your method but should not clutter the final figure. After confirming the construction is correct, darken the final sides of the figure neatly and carefully erase unnecessary arcs and construction lines. Do not erase the labels or tick marks. A neat darkened final figure is easier to grade and demonstrates control over instruments.

Writing steps and instrument names: Number the construction steps and begin each with an action verb (draw, place, join, measure). Mention which instrument you use in each step (ruler, compass, protractor, set-square) and give exact lengths or angles used. At the end add a brief check statement such as 'Measured AB = 5 cm, angle at A = 60°' to show verification.

Layout and spacing: Leave enough space on the page around the drawing so labels do not overlap. Keep the figure proportionate and centred in the available area. If multiple small constructions are required, separate them to avoid confusion. Practise neat constructions on scrap paper to improve speed without losing quality.