A geometrical construction means to draw geometrical figures, such as an angle, a circle, a triangle, a quadrilateral, and a polygon, etc.
We normally use all or some of the following instruments for drawing geometrical figures:
Q1. Construct an angle of 90º at the initial point of a given ray and justify the construction.
Ans: Steps of construction:
Thus, ∠AOF = 90º.
Justification:
∵ O is the centre of the semicircle and it is divided into 3 equal parts.
∴
∠BOC = ∠COD = ∠DOE
∵ Equal chords subtend equal angles at the centre
∴ ∠BOC + ∠COD + ∠DOE = 180º
∠BOC + ∠BOC + ∠BOC = 180º
3∠BOC = 180°
∴ ∠BOC = 60º
Similarly, ∠COD = 60º and ∠DOE = 60º
∵ OF is the bisector of ∠COD.
∴ = 30º
Now, ∠BOC + ∠COF = 60º + 30º
∠BOF = 90º or ∠AOF = 90º
Q2. Construct an angle of 45º at the initial point of a given ray and justify the construction.
Ans: Steps of construction:
Thus, ∠BOG = 45º or ∠AOG = 45º
Justification:
∵
∴ ∠BOC = ∠COD = ∠DOE
∵ Equal chords subtend equal angles at the centre
∴ ∠BOC + ∠COD + ∠DOE = 180º
∠BOC = 60º
∵ is the bisector of ∠BOC/
∴ .. (1)
Also, is the bisector of ∠COF.
∴ ... (2)
Adding (1) and (2), we get∠COF + ∠FOG = 30º + 15º = 45º
∠BOF + ∠FOG = 45º [∵ ∠COF = ∠BOF]
∠BOG = 45º
Q3. Construct the angles of the following measurements:
(a) 30º
(b)
(c) 15º
Ans:
(a) Angle of 30º
Steps of construction:
Thus, ∠BOD = 30º
(b) Angle of
Steps of construction:
Thus,
(c) Angle of 15º
Steps of construction:
Thus, ∠AOD = 15º
Q4. Construct the following angles and verify by measuring them by a protractor:
(a) 75º
(b) 105º
(c) 135º
Ans:
(a) Angle of 75º (Hint: 75º = 60º + 15º)
Steps of construction:
Thus, ∠BOQ = 60º + 15º = 75ºor ∠AOQ = 75º.
(b) Angle of 105º (Hint: 105º = 90º + 15º)
Steps of construction:
Thus, ∠AOQ = 105º
(c) Angle of 135º (Hint: 120º + 15º = 135º)
Steps of construction:
Thus, ∠POM = 135º.
Q5. Construct an equilateral triangle, given its side and justify the construction.
Ans: Let us construct an equilateral triangle, each of whose side = PQ
Steps of construction:
Thus, ΔOBC is the required equilateral triangle.
Justification:
∵ The are drawn with the same radius.
∴ ⇒
∵ Chords corresponding to equal arcs are equal.
∵ OC = OB = BC
∴ ΔOBC is an equilateral triangle.
1402 docs679 tests

1. What are the basic constructions in geometry? 
2. How do you construct a perpendicular bisector of a line segment? 
3. What is the purpose of constructing an angle bisector? 
4. How can you construct a perpendicular from a point to a line? 
5. Can you explain how to construct parallel lines? 
1402 docs679 tests


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