Introduction to Constructions: Bisectors, Angles & Triangles

Construction is a part of Geometry that helps us in understanding how we can draw shapes, angles, or lines accurately. There are various construction methods to precisely construct many geometric figures. These constructions are done by using a compass, ruler, and pencil. At a complex level, geometric constructions are done using high-level machinery and application systems.

Key Takeaways: Triangle, Angle Bisector, Perpendicular Bisector, line segment.

Also Read: Trigonometry Identities

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Construction of Angle Bisector

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Angle bisector is that line segment which bisects or divides an angle into two equal parts.

Steps of Construction:

• Take any radius of your choice and draw an arc taking B as the centre. Intersect the rays BA and BC with the arc, say at E and D, respectively.
• Now, take a radius of more than 1/2 DE, and take D and E as centres. Draw arcs to intersect each other, say at F.
• Join the points B and F, to form a ray BF, which is the required bisector of the ∠ABC.

Angle Bisector Construction

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Construction of a Perpendicular Bisector

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Perpendicular bisector is a line segment which divides a line into two equal parts.

Steps of Construction

• Draw a line AB of a certain length.
• Taking A and B as centers and a radius more than 1/2 AB, draw arcs on both the sides of the line segment AB, in such a way that they intersect each other at points P and Q.
• Let these points join to form the line PQ and intersect AB at point M.
• The line obtained, that is PMQ, is the required perpendicular bisector of AB.

Perpendicular Bisector Construction

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Constructing an Angle of the Measure 60°

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Angles between 0° and 90° are called an acute angle. To construct an acute angle of the measure 60°, the following steps are to be followed.

Steps of Construction

• Take any radius of your choice, and draw an arc taking A as the center of a circle that intersects the line AB, at a point D.
• Take the same radius as before, and draw an arc taking D as the center, in such a way that it intersects the previously drawn arc, say at a point E.
• Join the points A and E to form a ray AEC. 
• So, ∠CAB is the required angle of 60°.

Construction of 60° angle

Also Read: Surface Areas and Volumes

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Construction of Triangle

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Triangle is a three-sided polygon having three vertices. The sum of all interior angles of any triangle is 180°. The following steps are to be followed to construct a right angled triangle.

Given: base, base angle, and the sum of the other two sides

Steps of Construction:

(i) Draw the base BC and at point B, makes an angle, say ∠XBC equal to the given angle.

(ii) Cut a line segment BD equal to AB + AC from the ray BX.

(iii) Join DC and make an angle ∠DCY equal to ∠BDC.

(iv) Let CY intersect BX at A. 

Triangle ABC is the required triangle.

Given: Base, Base angle and the difference of other two sides

Case (I): Let AB > AC, i.e., AB-AC is given.

Steps of Construction

• Draw the base BC and at point, B makes an angle, say ∠XBC equal to the given angle.
• Cut the line segment BD equal to AB – AC from ray BX.
• Join DC and draw the perpendicular bisector, say PQ of DC.
• Let it intersect BX at a point A. Join AC.

Then, triangle ABC is the required triangle.

Case (II): Let AB < AC i.e., AC – AB be given.

Steps of Construction

• Draw the base BC and at point, B makes an angle, say ∠XBC equal to the given angle.
• Cutline segment BD equal to AC – AB from the line BX extended on the opposite side of line segment BC.
• Join DC and draw the perpendicular bisector, say PQ of DC.
• Let PQ intersect BX at A. Join AC.

Then, ABC is the required triangle.

Given: Perimeter and the two base angles

Steps of Construction

• Draw a line segment, say XY equal to BC + CA + AB.
• Make angles LXY equal to ∠B and MYX equal to ∠C.
• Bisect ∠LYX and ∠MYX. Let these bisectors intersect a point A.
• Draw perpendicular bisectors PQ of AX and RS of AY.
• Let PQ intersect XY at B and RS intersect XY at C. Join AB and AC.

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Things to Remember

• There are various construction methods to precisely construct many geometric figures. 
• These constructions are done by using a compass, ruler, and pencil.
• Angle bisector is that line segment which bisects or divides an angle into two equal parts.
• Perpendicular bisector is a line segment which divides a line into two equal parts.
• Angles between 0° and 90° are called an acute angle.

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