Why do we learn constructions? Suppose a building needs to be constructed. One needs to know the accurate map of the building. Construction means to draw the lines and angles accurately. Also to draw the road maps, learning the basic constructions is of great importance. Let us now learn about the basic constructions in geometry.
1. Construction of a bisector of a given angle
- With A as a centre and using compasses, draw an arc that cuts both rays of A.
- Label the points of intersection as B and C.
- Now with B as a centre, draw (in the interior of A) an arc whose radius is more than half the length BC.
- With the same radius and with C as a centre, draw another arc in the interior of A.
- Let the two arcs intersect at D
- So we get AD as the required bisector of A.
2. Construction of the perpendicular bisector of a given line segment.
- Draw a line segment AB
- Now take A as a centre, using compasses, draw a circle.
- The radius of your circle should be more than half the length of AB.
- With the same radius and with B as a centre, draw another circle using compasses.
- Let it cut the previous circle at C and D.
- Join CD so it cuts AB at O.
- O is the midpoint of AB. Also, COA and COB are right angles.
- Therefore, the CD is the perpendicular bisector of AB.
3. Construction of 60° Angle
- Take a ruler and draw a line l and make a point O on it. Take a compass and put its one end at point O and draw an arc with any convenient radius.
- We take the centre O and draw an arc. We get the point A.
- Now take the compass and with the same orientation taking the point A as the centre draw an arc that passes through O. Draw an arc such that it intersects the existing arc.
- Here OA and OB are nothing but the radius which is equal in length.
- So actually we are trying to draw an equilateral triangle which means the angle we see in the above construction is 60 degree.
- We get ∠BOA which measures 60 degrees.
Solved Examples for You on Basic Constructions
Question 1: To construct a perpendicular to a line L from a point P outside the line, steps are given in the jumbled form. Identify the second step from the following.
1)Draw line PQ
2)Draw a line L and consider point P outside the line.
3)Take P as a centre, draw 2 arcs on line L and name it as points A and B respectively.
4)Taking A and B as a centre one by one and keeping the same distance in compass, draw the arcs on another side of the plane.The point where these arcs intersect name that point as Q
Solution: B is the correct option. 3 is the second step amongst all.
Question 2: While constructing a parallel line to a given line, we ______.
- Copy a segment
- Bisect a segment
- Copy an angle
- Construct a perpendicular
Solution: C is the correct option. While constructing a parallel line to a given line, we copy an angle.