Chapter Notes: Practical Geometry

Lines That Don't Meet

Method of construction of a line parallel to a given line, using only a sheet of paper

• Take a piece of paper.
• Fold it in half and unfold the line l. Mark a point A on paper outside l.
• Fold the paper perpendicular to the line such that this perpendicular passes through A. Name the perpendicular AN.
• Make a fold perpendicular to AN through point A. Name the new perpendicular line as m.
• Now,  l || m.

Steps of construction of a line parallel to a given line

• Take a line l and a point A outside l.
• Take any point B on l and join it to A.
  
• With B as the centre and a convenient radius, cut an arc on l at C and BA at D.
  
• With A as the centre and same radius as in Step 3, cut an arc EF to cut AB at G.
  
• Measure the arc length CD by placing pointed tip of the compass at C and pencil tip opening at D.
• With this opening, keep G as centre and draw an arc to cut arc EF at H
  
• Join AH to draw a line m
  
• ∠ABC and ∠BAH are alternate interior angles. Therefore, m || l
• To know more about Construction of a Parallel Line.

Let's Build Triangles

Classification of triangles based on sides and angles

Triangles can be classified based on their:

1. Sides

• Equilateral triangle: All three sides are equal in measure.
• Isosceles triangle: Two sides have equal measure.
• Scalene triangle: All three sides have different measures.

2. Angles

• Acute triangle: All angles measure less than 900.
• Obtuse triangle: One angle is greater than 900.
• Right triangle: One angle is 900.

Important properties of triangles

• The exterior angle is equal to the sum of interior opposite angles.
• The sum of all interior angles is 180°
• Sum of the lengths of any two sides is greater than the length of the third side.
• Pythagoras theorem: In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
  
• Triangles can be constructed if any of the following measurements are given
  • Three sides.
  • Two sides and an angle between them.
  • Two angles and a side between them.
  • The hypotenuse and a leg in case of a right-angled triangle.

Construction of triangles given a criterion are listed below:

1.  Construction of a triangle with SSS criterion.

Construct a triangle ABC, given that AB = 4.5 cm, BC = 5 cm and AC = 6 cm.

Steps:

• Make a rough sketch for your reference
• Draw a line segment BC = 5 cm
  
• With B as centre, draw an arc of radius 4.5 cm
  
• With C as centre, draw an arc of radius 6 cm and cut the previous arc
  
• Mark the point of intersection of arcs as A. Join AB and AC. ΔABC is now ready

Note: SSS congruency rule: If three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent

2. Construction of a triangle with SAS criterion
Construct ΔPQR with QR = 7.5 cm, PQ = 5 cm and ∠Q = 600.

Steps:

• Make a rough sketch for your reference
• Draw a line segment QR = 7.5 cm
  
• At Q, draw QX making 600 with QR
  
• With Q as centre, draw an arc of radius 5 cm. It cuts QX at P.
  
• Join AB. ΔPQR is now ready
  

3. Construction of a triangle with ASA criterion

Construct ΔXYZ with ∠X = 300, ∠Y = 1000 and XY = 5.8 cm.

Steps:

• Make a rough sketch for your reference
• Draw XY = 5.8cm
  
• At X, draw a ray XP making an angle of 300 with AB.
  
• At Y, draw a ray YQ making an angle of 1000 with XY.
  
• The point of intersection of the two rays is Z.
• ΔXYZ is now completed
  

4. Construction of a triangle with RHS criterion
Construct ΔLMN, where ∠M = 900, MN = 8cm and LN = 10 cm.
Steps:

• Make a rough sketch for your reference
• Draw MN = 8 cm
  
• At M, draw MX ⊥ MN.

  

• With N as centre, draw an arc of radius 10 cm to cut MX at L
  

  
  

• Join LN.

• ΔLMN is now completed

Basics of Practical Geometry

Introduction to Constructions of basic figures

Basic constructions:

• To draw a line segment of given length
• a line perpendicular to a given line segment
• an angle
• an angle bisector
• a circle

In geometry, you have already studied basic constructions.

• To recap, some of them are:
  • Drawing a line segment of given length
  • Drawing a line perpendicular to a given line segment.
  • Angles
  • Angle bisectors
  • Circles
• Tools used for simple constructions are ruler, protractor and a compass.