Essence of Geometrical Constructions
Geometry is a concept that deals with lines, angles, shapes of objects, sizes and dimensions. To represent shapes on a paper we need to draw them accurately with the help of tools like rulers, protractors or compasses. The simplest construction is that of a line. We start with a point and extend it slowly to draw a line. A line can have a fixed measurement or an infinite length. Similarly, angles, circles and other shapes can be constructed with the correct procedure and the given dimensions.
Geometrical Construction Definition
Geometrical construction means drawing lines, line segments, shapes, circles and other figures accurately using a ruler, a compass, or a protractor. Most of the geometrical figures involve drawing a line segment, drawing parallel and perpendicular lines, perpendicular bisectors, circles, and even drawing tangents to circles.
Basic Geometric Constructions
The following section explains the basic geometric constructions for angles and circles.
Angles
There are three types of angles. The following lines describe how these types of angles can be drawn with the help of a protractor and a ruler.
Acute Angle: An angle whose measure is less than 90° is called an acute angle. Let us draw an acute angle of 40° with the help of the following steps.
- Step 1: Draw a straight line with a ruler and mark the end points as A and B.
- Step 2: Place the protractor on point 'A' such that the point coincides with the centre of the protractor.
- Step 3: Now, considering the bottom scale of the protractor, identify 40° and mark it 'C'. Join point 'C' to point 'A' to form an acute angle CAB = 40°
The construction of an acute angle is shown in the figure below.
Obtuse Angle: An angle whose measure is greater than 90° is called an obtuse angle. Let us draw an acute angle of 110° with the help of the following steps.
- Step 1: Draw a straight line with a ruler and mark the end points as Q and R.
- Step 2: Place the protractor on point 'Q' such that the point coincides with the centre of the protractor.
- Step 3: Now, considering the bottom scale of the protractor, identify 110° and mark it 'P'. Join point 'P' to point 'Q' to form an obtuse angle PQR = 110°
The construction of an obtuse angle is shown in the figure below.
Right Angle: A right angle measures exactly 90° and we can construct a right angle with the help of the following steps.
- Step 1: Draw a straight line with a ruler and mark the end points as A and B.
- Step 2: Place the protractor on point 'A' such that the point coincides with the centre of the protractor.
- Step 3: Now, considering the bottom scale of the protractor, identify 90° and mark it 'C'. Join point 'C' to point 'A' to form a right angle CAB = 90°.
The construction of a right angle is shown in the figure below.
Circles
Some important terms related to circles are discussed below.
- Circumference: Circumference is defined as the boundary of a circle.
- Radius: Radius is measured as the distance between the center of the circle to any point on the circumference.
- Diameter: Diameter is a line segment passing through the center of the circle, whose endpoints touch the circumference of the circle.
- Chord: Chord is the line segment joining any two points on the circumference of a circle.
Follow the steps given below to draw a circle of radius 5 units using a compass.
- Step 1: Using a ruler, set the required width in the compass to 5 units.
- Step 2: Place the pointed tip of the compass at a point considering it to be the center of the circle and rotate it fully to draw a circle.
The circle with radius 5 units will look as shown in the following figure.
Terms and Definitions Used in Geometrical Constructions
- Bisect: In geometrical constructions, bisect means dividing into two equal parts. We can bisect a line, an angle and even shapes.
- Parallel : When two lines are drawn on a plane that do not meet each other, they are called as parallel lines.
- Perpendicular: When two lines drawn meet each other at a right angle or 90°, they are called perpendicular lines.
- Tangent: Tangent is defined as a straight line that touches a curve at a point.
- Inscribed: When a polygon is fully drawn inside a circle, we can say that the polygon is inscribed in the circle. It can also be said that the circle is circumscribed about the polygon.
- Circumscribed: When a polygon surrounds another geometrical figure (say, a circle) such that all the vertices of the inner figure touch the sides of the outer polygon, then the polygon is said to be circumscribed about the circle.
The following figures describe each of the geometric terms discussed above.
Construction of a Triangle
Now let us learn how to construct a triangle when its three sides are given. Let us construct a triangle ABC where AB = 5 units, BC = 7 units and AC = 4 units. We will need a ruler, a compass and a pencil. The following steps show the way to construct a triangle.
- Step 1: Draw the longest given line segment measuring 7 units with the help of a ruler and mark it as BC.
- Step 2: Use a ruler and measure 5 units with the compass and draw an arc above the line with B as the center.
- Step 3: Then measure 4 units with the compass and draw an arc above the line with C as the center such that it intersects the arc drawn in step 2 and mark the intersecting point as 'A'.
- Step 4: Join the points AB and AC.
The following figure shows the construction of a triangle with the given measurements.
Topics Related to Essence of Geometric Constructions
- Geometric Construction
- Basic Triangle Constructions
- Geometrical Proofs
- Circles
- Constructing Circles
- Radius
- Radius of Circle Calculator
Solved Examples
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Example 1: Draw a perpendicular bisector of a line segment of length 8 units.
Solution:Step 1: Draw a line measuring 8 units with a ruler and mark it as AB.
Step 2: Using a compass, take A as the center and the radius more than half of the length of AB (more than 4 units), draw arcs both above and below the line AB.
Step 3: Now, keeping B as the centre and the same radius, draw arcs both above and below the line AB such that the arcs intersect.
Step 4: Take a ruler and join the points of intersection of the arcs above and below the line. This line is the perpendicular bisector of line AB.
The construction of a perpendicular bisector is shown below.
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Example 2: Construct an angle bisector for the given angle ABC.
Solution:
Step 1: Using a compass, take point B as the center and draw an arc on line BC and mark it as Q. Draw another arc on line AB and mark it as P.
Step 2: Now with the pointed edge of the compass on P, make an arc within the lines AB and BC. Do the same with Q as the center and make an arc such that the second arc intersects the first arc.
Step 3: Draw a line from the point of intersection to the point B.
Step 4: This line is called the angle bisector of angle ABC.
The construction of the angle bisector is shown in the following figure.
FAQs on Essence of Geometric Constructions
What is Geometric Construction?
Geometric construction means to draw or construct geometric figures which involve lines, angles, shapes. We usually use a compass, a ruler, a protractor and a pencil to construct these figures.
What are the Tools Used in Geometric Construction?
A ruler, compass, protractor, set squares and a divider are the basic tools used for geometrical construction.
What are the Four Basic Geometric Constructions?
In geometry, the four basic constructions are:
- Drawing a line with two given points.
- Finding the midpoint of a line segment.
- Constructing a perpendicular bisector.
- Drawing an angle bisector.
What are the Basic Terms used in Geometric Constructions?
The basic terms used in geometric constructions are as follows:
- Point: A point describes a position or a location. There is no specific size for a point.
- Line: A line is a figure with only length and no width. It extends infinitely in both directions.
- Line Segment: Line segment is a part of a line with a definite length and two endpoints.
- Radius: Radius is measured as the distance between the center of the circle to any point on the circumference.
- Bisect: Bisect means exactly dividing into two equal parts. A line, angle or any shape can be bisected.
- Arc: A curved part is generally described as an arc. Arc is also a part of the circle.
What Does it Mean to Bisect a Segment?
Bisecting a line segment means to divide it equally into two parts. A bisector is defined as a line segment that passes through the midpoint of the given line segment.