Chapter 10 — Practical Geometry

Practical Geometry studies how to make accurate geometric drawings using simple instruments. It connects theoretical geometry (lines, angles, triangles, circles) with hands-on construction, so you can reproduce, measure and verify geometric figures precisely.

Key aims: learn the correct use of drawing tools, follow step-by-step constructions, and understand how measurement, scale and accuracy affect a drawing. Good technique ensures constructions are reproducible and error is minimized.

Common tools and their purposes:

• Ruler (scale): for drawing straight lines and measuring lengths in millimetres and centimetres.
• Compass: for drawing circles and transferring distances (arcs, equal segments).
• Protractor: for measuring and constructing angles in degrees.
• Set-squares (45° and 30°–60°): for drawing perpendiculars and fixed-angle lines.
• Divider: for stepping off equal lengths along a line.
• Sharp pencil, eraser, sharpener: for clear, thin lines and corrections.
• Drawing board or clean flat surface: for stability and accuracy.

Basic rules and tips: use a sharp HB pencil for clear lines; hold the instruments correctly (compass point steady, ruler flush against paper); make light construction lines first, darken final lines; check measurements twice; record the units and any scale used.

Accuracy concepts: least count (smallest division on the ruler, typically 1 mm), propagation of error (small measurement errors can change constructions), and scale (to draw large or small objects proportionally).

What are Basic Constructions?
Basic constructions are ways to draw precise geometric figures (lines, angles, perpendiculars, parallels, bisectors, etc.) using only a ruler (unmarked straightedge) and a compass. These constructions rely on simple geometric properties and are the foundation of Practical Geometry in Class 7.

Common tools: ruler (unmarked), compass, pencil, eraser.

Important ideas used

• If a point lies on the perpendicular bisector of segment AB, it is equidistant from A and B.
• If a ray bisects an angle, every point on that ray is equidistant from the sides of the angle (locally: the bisector divides the angle into two equal angles).
• Parallel lines have equal corresponding/alternate interior angles when cut by a transversal. Two successive perpendiculars give a parallel (perpendicular to the same line are parallel to each other).

Standard constructions (step-by-step idea)

• Copy a given line segment AB of length l: Put compass at A to B (set distance AB). On a ray from new point X, mark off that compass width to get Y so XY = AB.
• Perpendicular bisector of segment AB: With compass radius > half AB, draw arcs from A and B to produce two intersection points above and below the segment. Join the two intersection points; that line is the perpendicular bisector and meets AB at its midpoint.
• Bisect angle ∠AOB: With centre O and any radius draw arc cutting OA and OB at P and Q. With equal radii draw arcs from P and Q to meet at R. Join O to R. OR is the angle bisector (∠AOR = ∠ROB).
• Perpendicular to a line through a point on the line (point P on line l): Mark two points A and B on line l equidistant from P. From A and B draw arcs of same radius on one side so they meet at Q. Join P to Q. PQ is perpendicular to l at P.
• Perpendicular to a line through a point not on the line (point P not on l): With centre P draw an arc cutting l at A and B. From A and B draw equal arcs that intersect at Q. Join P to Q. PQ is perpendicular to l.
• Construct a line parallel to a given line l through point P: Two common methods:
  • Copy an angle: draw a transversal through P making any angle with l, copy the corresponding angle at P so the corresponding/alternate interior angles are equal — the new line through P is parallel to l.
  • Double-perpendicular method: draw a perpendicular to l, then draw a perpendicular to that through P. The result is parallel to l.
• Construct 60° (equilateral triangle method): Draw segment AB. With centre A and radius AB draw an arc; with centre B and same radius draw another arc; let them meet at C. Join AC and BC. Triangle ABC is equilateral; each angle = 60°.

How to present a construction answer (recommended format):

1. Given: (state given data)
2. To construct: (what you will draw)
3. Steps of construction: (numbered step-by-step compass & straightedge operations)
4. Reason: (short justification using geometric facts)

Why these matter (intuition and use): Constructions produce exact figures without measuring angles numerically. The repeated use of equal compass radii and intersection points encodes equality and symmetry — e.g., intersections of equal arcs produce midpoints, perpendicular lines, or equal angles.

Definition: Two lines are perpendicular if they meet at a right angle (90°). In practical geometry we construct perpendicular lines using a ruler and compass. Common constructions include: (1) a perpendicular through a point on a line, (2) a perpendicular from a point not on a line, and (3) the perpendicular bisector of a segment.

Tools: ruler (straightedge) and compass.

1. Perpendicular through a point P on a given line l

1. With centre P and any convenient radius, draw an arc that cuts line l at two points A and B equidistant from P.
2. With centres A and B and a radius larger than half AB, draw two arcs on one side of l (and optionally on the other side) so that they intersect at points X and Y.
3. Join X and Y with the ruler. Line XY is the perpendicular to l at P (it passes through P and makes 90° with l).

2. Perpendicular from a point P not on line l

1. With centre P and a radius large enough to cut l, draw an arc that meets l at A and B (so PA = PB).
2. Construct the perpendicular bisector of AB (with centres A and B draw arcs that intersect at two points; join those points).
3. The perpendicular bisector of AB passes through P and is perpendicular to l. Join P along that direction to get the perpendicular to l through P.

3. Perpendicular bisector of a segment AB

1. With centres A and B and radius greater than half AB, draw arcs that intersect in two points (one on each side of AB).
2. Join the intersection points. This line is the perpendicular bisector: it meets AB at its midpoint M and at right angle.

Key properties:

• A perpendicular makes an angle of 90° with the line it meets.
• The perpendicular bisector of segment AB meets AB at its midpoint and every point on the bisector is equidistant from A and B.
• If two lines are perpendicular, the product of their slopes (in coordinate form) is -1 (when slopes are defined).

These constructions rely on the compass property: equal radii from the same centre give equal distances, and the perpendicular bisector is the locus of points equidistant from two given points.

Definition: An angle bisector is a ray or line that divides an angle into two equal angles. If ∠AOB is bisected by ray OC, then ∠AOC = ∠COB.

Key properties:

• Each part equals half of the original angle: if ∠AOB = 2θ, then each bisected angle = θ.
• Every point on the angle bisector is equidistant from the two sides of the angle. (If P lies on the bisector of ∠AOB, the perpendicular distances from P to OA and OB are equal.)
• In a triangle, an internal angle bisector divides the opposite side in the ratio of the adjacent sides (Angle Bisector Theorem): if AD is the bisector of ∠A in triangle ABC meeting BC at D, then BD/DC = AB/AC.

How to construct an angle bisector (compass and ruler):

1. Given vertex O and rays OA and OB, place compass at O and draw an arc that cuts both rays at points P (on OA) and Q (on OB).
2. With the same compass width, draw arcs centered at P and Q so that the two arcs intersect at point R (on the interior of the angle).
3. Draw ray OR. OR is the bisector of ∠AOB because it creates equal arcs and hence equal angles.

Brief reasoning for equidistance property: Drop perpendiculars from a point P on the bisector to the sides OA and OB. The two right triangles formed are congruent by AAS (equal angle, common perpendicular length, and equal acute angle), so the perpendicular distances are equal.

Introduction
Construction of triangles is the process of making a triangle accurately using only a ruler (scale), a compass and sometimes a protractor. Certain sets of given elements (sides and/or angles) determine a unique triangle. These are called the basic cases of triangle construction.

Tools required

• Ruler (with scale)
• Compass
• Protractor (for angle constructions)
• Pencil and eraser

Basic cases (with steps and reasons)

1. SSS (Three sides given)

Given three sides a, b and c (lengths), construct triangle ABC with AB = c, BC = a and CA = b.

1. Draw a base AB of length c using the ruler.
2. With centre A and radius b, draw an arc.
3. With centre B and radius a, draw another arc. The intersection point of the two arcs is C.
4. Join C to A and C to B.

Reason: C is the common point whose distances from A and B are fixed, so the intersection of two circles gives the unique position (up to reflection) of C. Check triangle inequality a + b > c, etc., before construction—if it fails, a triangle cannot be formed.

2. SAS (Two sides and included angle given)

Given sides AB = c, AC = b and the included angle ∠BAC = θ, construct triangle ABC.

1. Draw side AB = c as the base.
2. At A, construct angle ∠BAC = θ using a protractor or compass construction.
3. On the ray making angle θ with AB, measure and mark AC = b using the compass from A to get point C.
4. Join C to B.

Reason: fixing two sides with the included angle fixes the triangle uniquely up to congruence (SAS congruence).

3. ASA (Two angles and included side given)

Given side BC = a and the angles ∠ABC = β and ∠BCA = γ (angles at B and C), construct triangle ABC.

1. Draw base BC = a.
2. At B construct angle β and at C construct angle γ using a protractor or compass method.
3. The two rays from B and C meet at A. Join A to B and A to C.

Reason: Two angles and the included side fix the remaining vertex because the sum of interior angles is 180°, so the rays intersect in one point.

4. (Related) AAS or SAA (Two angles and a non-included side)

If two angles and any one side are given, you can first find the third angle (180° − given angles) and then construct as ASA.

5. RHS (Right angle, Hypotenuse and one Side)

For a right triangle: given the hypotenuse and one leg or given hypotenuse and an acute angle, construct by placing the hypotenuse as base and using right-angle or circle properties (Thales theorem) to locate the other vertex.

Verification and tips

• Check the triangle inequality before SSS construction: any two sides must sum to more than the third.
• Label points as you construct to avoid confusion.
• Use sharp pencil and steady compass settings for accuracy; redraw arcs if compass slips.
• After construction you can verify angles add to 180° and side lengths match the given values within measurement tolerance.

Why these cases work (brief)

These sets are sufficient because they match standard congruence rules: SSS, SAS and ASA (and AAS/RHS) guarantee that only one triangle (up to congruence or mirror image) can be formed from the given data.

A right-angled triangle is a triangle in which one angle is 90°. Practical construction uses a ruler and compass and basic geometric facts (Thales' theorem, perpendicular lines, angle construction). Two common construction cases studied in Class 7 are:

• Case 1 – Hypotenuse and one acute angle given (H, A):

  Let the required right triangle be ABC with right angle at C, hypotenuse AB = given length and angle ∠A = given acute angle. By Thales' theorem every point on the circle having AB as diameter gives a right angle at that point. So construct segment AB of given length, draw the circle with AB as diameter, then from A construct the given angle ∠A (a ray from A). The intersection of that ray with the circle is point C. Join C to A and B to complete triangle ABC.

• Case 2 – Base and acute angle at one end given (B, ∠A):

  Let AB be the given base and ∠A = given acute angle; suppose the triangle is right-angled at C. Then ∠B = 90° − ∠A. Construct AB, at A construct angle ∠A (ray for AC), and at B construct angle 90° − ∠A (ray for BC). The intersection of the two rays is point C. Join AC and BC to get the right triangle.

Key geometric tools and facts used:

• Thales' theorem: angle in a semicircle is a right angle (useful when hypotenuse is given).
• Construction of angles and perpendiculars with ruler and compass.
• Complementary angles: if one acute angle is given, the other acute angle in a right triangle equals 90° minus the given angle.

General construction steps (summary):

1. Draw the given segment (hypotenuse or base) with a ruler.
2. Using a protractor/compass construct the required ray(s): the given acute angle at an endpoint and/or the complementary angle at the other endpoint.
3. If hypotenuse is given, draw the circle having that segment as diameter and find intersection with the ray to locate the right vertex.
4. Join the three vertices to complete the triangle and verify the right angle with a constructed perpendicular.

Tips for accuracy: use a sharp pencil, steady compass, and verify with a right-angle construction (e.g., construct a perpendicular) or measure the angle with a protractor.

What this topic means
Beyond the standard construction cases (SSS, SAS, ASA, RHS) there are many other kinds of construction problems that appear in Class 7 Practical Geometry. These are problems where you are given combinations such as a base and the sum (or difference) of the other two sides, a base and an altitude (height), a base and a median, base angles together with a constraint on the two sides, or the perimeter instead of individual sides. The methods used in these problems extend the basic tools: ruler, compass, straightedge and the ideas of loci and simple geometric transformations.

Main idea / strategies

• Reduce to a known case: Transform the given data so the problem becomes one of SSS, SAS, ASA or RHS. For example, use an auxiliary point or reflect a point to convert a "sum of two sides" condition into a single distance condition.
• Locus method: Many conditions describe a locus (set) of possible positions of a vertex. For example, the set of points whose distances from two fixed points have a constant sum is an ellipse (conceptually). In constructions you often replace the curve by an equivalent construction trick.
• Use auxiliary lines and reflections: For a condition like AB + AC = constant, you can create an auxiliary point such that one distance equals that constant (by translating/reflecting one side) and then solve by simple circle intersections.
• Use known triangle elements: If a median, altitude or angle is given along with a base, construct the median/altitude line first; intersection with appropriate circles or rays gives the required vertex.

Important points to remember

• Always mark the given lengths and angles clearly and draw only what is necessary for the construction.
• Check the triangle inequality: any side must be less than the sum of the other two sides and greater than their difference; otherwise a triangle cannot be constructed.
• When an unusual condition (sum/difference/perimeter) is given, look for an auxiliary construction that converts it to a standard intersection-of-circles or intersection-of-lines problem.

Sample conceptual constructions (types)

• Given base BC and AB + AC = s: Conceptually you want points A such that distance to B plus distance to C is s. In advanced geometry this locus is an ellipse. In elementary constructions you avoid drawing the ellipse by introducing an auxiliary point or translating one of the fixed points so the condition becomes a single distance (then use circle intersections).
• Given base BC and |AB − AC| = d (difference): The locus of points with constant difference of distances to B and C is a branch of a hyperbola. In practice construct an auxiliary point on BC or use reflection to turn the difference into one distance, then use circles.
• Given base BC and median from A: Construct the midpoint M of BC (with compass and straightedge). If the length of the median AM is given, draw a circle with centre M and radius = given median; intersections with the perpendicular line(s) through M or with appropriate rays give vertex A.
• Given base BC and altitude (height) from A: Draw the perpendicular to BC at the point where the altitude meets BC (if that foot is given) or use the given height to draw a line parallel/perpendicular at correct distance, then locate A where this line meets the locus defined by other given data (like an angle or side length).

How to check your construction

• Measure the constructed sides and angles with a ruler/compass and protractor to confirm they match the given data.
• Verify the triangle inequality and any other numerical constraints (sum/difference/perimeter).

Definition: Parallel lines are two straight lines in the same plane that never meet, no matter how far they are extended. They are always the same distance apart.

Key properties: If two lines are parallel then (i) corresponding angles formed by a transversal are equal, (ii) alternate interior angles are equal, and (iii) consecutive interior angles sum to 180°.

Practical construction idea (compass and ruler) — To draw a line through a point P parallel to a given line l:

1. Let l be the given line and P be the point not on l.
2. Draw any transversal joining P to line l; let it meet l at Q (draw line PQ).
3. At Q, we will copy the angle made by PQ with line l at point P. With centre Q and a convenient radius, draw an arc that cuts the transversal PQ at B and the line l at A.
4. With the same radius, draw an arc with centre P; this arc will cut the region where the new line should go at a point B'.
5. Measure the distance AB using the compass. With centre B' and radius AB draw an arc to meet the arc drawn in step 4 at C.
6. Join P to C. The line PC is parallel to l (because the angle copied at P is equal to the angle at Q, giving equal alternate interior/corresponding angles).

Alternate quick method (set-square and ruler): Place one set-square along the given line l, slide the second set-square along the first so that its edge passes through the point P, and draw the line along that edge — this produces a line parallel to l.

Why it works: By copying the angle that the transversal makes with the given line at Q to the point P, we ensure the corresponding/alternate interior angles are equal; equal corresponding or alternate interior angles imply lines are parallel.

Notes: The same angle-copying idea works to construct a line parallel to any given line passing through any given point (on or off the line). In coordinate geometry, parallel lines have equal slopes.

What the topic means
In practical geometry, after constructing a figure (lines, angles, triangles, bisectors), we must verify that the construction meets the given conditions, understand sources of inaccuracy, and learn where these constructions are used in real life (applications).

Verification
Verification means checking the required properties of a construction by measurement or by using geometric properties. Typical verification checks:

• Perpendicular lines: check with a set-square or measure the angle as 90° with a protractor.
• Perpendicular bisector: choose any point on the bisector and verify it is equidistant from the endpoints of the segment (use a compass or ruler).
• Angle bisector: measure the two angles made with the sides using a protractor; they should be equal. Alternatively, a point on an angle bisector is equidistant from the sides of the angle.
• Triangle constructions (SSS, SAS, ASA): measure the three sides or the given sides and included angle to ensure they match the given data; check that the sum of interior angles = 180° as a consistency test.
• Right triangle: verify using the Pythagorean relation a² + b² = c² for the sides, if appropriate.

Accuracy
Accuracy refers to how close a measured or constructed value is to the true or required value. Sources of inaccuracy and ways to reduce them:

• Instrument limits (least count): a typical ruler reads to 1 mm, a protractor to 1°; finer instruments give better precision.
• Human errors: parallax when reading, pressing compass too hard, blunt pencil tip. Reduce by steadying instruments, using sharp pencils and reading at eye level.
• Repeat measurements and take averages to reduce random errors.
• Work on larger scale drawings when possible — the same absolute instrument error becomes a smaller relative error.
• Report measurement uncertainty: use units and, if needed, state possible error (for example 12.3 cm ± 0.1 cm).

Applications
Geometric verification and accuracy are used in many practical areas:

• Carpentry and construction: ensuring walls meet at right angles, making precise joins and frames.
• Engineering and technical drawing: accurate plans and blueprints depend on correct constructions and verification.
• Land surveying and mapping: locating midpoints, bisectors and measuring angles accurately for boundary marking.
• Tailoring and pattern making: accurate bisectors and measurements for symmetry and fit.
• Navigation and robotics: angle and distance constructions used in path planning and localization.

General verification procedure (stepwise)

1. Identify what must be true (e.g., AB = AC, angle B = angle C, line l ⟂ m).
2. Choose a verification tool (ruler, protractor, compass, set-square).
3. Measure/check the relevant quantities (angles, lengths, equal distances) carefully, preferably more than once.
4. Compare measurements with required values; accept if within instrument limits or stated tolerance.
5. If not acceptable, revise the construction and repeat verification.