## Geometric Constructions

Geometric construction is a process of constructing geometric figures using geometric tools such as a straightedge (ruler), a compass, and a pencil. This is the purest form of geometric constructions as no numbers are involved. The geometric tool such as compass is used to construct arc and circles and mark off equal length whereas the straightedge (ruler) is used to draw the line segmentsand measure their lengths.

In Geometric Construction basic, we will discuss different types of constructions such as the construction of line segments, copy of line segments, construction of angle, and angle bisector.

### Construction of Line Segment

A line segment is bounded by two fixed or definite ending points or we can say the line segment is a part of the line that joins two distinct points. The length of the line segment is measured in centimeters (cm), millimeters (mm), or by other conventional units such as feet or inches. Let us assume, we need to construct a line segment AB of length 5 cm. Look at the following steps to construct a line segment of 5 cm.

Step 1: Draw a line of any length, Mark a point A on the line, and consider it as a starting point of line segment.

Step 2: With the help of a ruler, set the pointer of the compass 5 cm apart from the pencil’s lead.

Step 3: Place the pointer of the compass at point A and draw an arc on the line with a pencil.

Step 4: Mark the point where the arc and line intersect at B

Hence, AB is the required length of 5 cm.

### Copy of Line Segment

We can construct a line segment using a ruler, compass, and pencil. Construction of a copy of the line segment can also be done using the above-mentioned tools. However, a line segment cannot be accurately constructed only using a ruler. Hence, a combination of ruler and compass is a better approach to construct a copy of the line segment. For constructing a copy of a line segment, it is necessary to draw a straight line for the given measure to start the procedure for copying the line segment. Such a line is known as a reference line. Let us learn to construct a copy of line segment XY using a ruler and compass.

Step 1: Draw a line segment XY of any length.

Step 2: Placing the pointer of the compass on the point X and the pointer of the pencil on point Y gives you the length of XY.

Step 3: Now, draw a straight line, say m using a ruler.

Step 4: Mark a point A on line ‘m’

Step 5: Taking A as center and taking the radius of length XY, draw an arc that cuts the line ‘m’ at the point, say B.

Hence, you can see the length of the line segment AB will be the same as that of the line segment XY ( if you don't change the radius of a compass).

### Construction of Angle

The construction of angle is an important part of Geometry as this knowledge enables us to construct other geometrical figures such as triangles. Construction of the angle of a given measurement, as well as unknown measurement, can be done using different geometric tools such as compass, protractor, and ruler. Here, we will discuss how to construct angles of a given measure such as 30º, 45º, 90º,120º, etc.

### How to Construct Angle Bisector`

The angle bisector is a line or ray that bisects the angle into two equal parts. Every angle has an angle bisector. You can easily construct an angle bisector using a ruler, compass, and pencil. Give ∠PQR, the steps to construct its bisector are:

Step 1: Place the pointer of the compass at point Q and draw an arc that cuts the two arms of an angle at two distinct points.

Step 2: From the points where the first arc cut at PQ, draw an arc arc towards the interior of the angle.

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Step 3: Using the same radius, draw an arc from the point where the first arc cut QR, cutting the previous arc drawn in step 2.

Step 4: Using a ruler, draw a line from point Q, where the arc intersects.

Hence, the line that is drawn through point Q represents the bisector of ∠PQR.

### How to Construct A 120 Degree Angle?

A 120-degree angle is double a 60-degree angle. Following are the steps to construct a 120-degree angle.

Step 1: Draw a line segment AB. Mark a left endpoint as O and a right endpoint as B.

Step 2: Taking O as center, and with a convenient radius, draw an arc that meets the line OB at say, P

Step 3: Taking P as center and with the same radius, draw an arc that cuts the first arc at a point, say Q.

Step 4: Taking Q as the center and with the same radius, draw an arc that cuts the first arc at a point, say A.

Step 5: Draw a line from point O through A.

Hence, we get the required angle i.e. ∠AOB = 120º.

### How to Construct A 60 Degree Angle?

A 60-degree angle is often used in geometric exercises, as it facilitates the constriction of several other angles of different measures. Following are the steps to construct a 60-degree angle.

Step 1: Draw a line segment of any length using a ruler. Mark the left endpoint of the line segment as 0 and the right endpoint as B.

Step 2: Take a compass and open it up to a convenient radius. Draw an arc from point O which meets the line OB at P.

Step 3: Taking P as center and with the same radius, draw an arc that passes through O and intersect the previous arc at a point, say A.

Step 4: Draw a line from O through A.

Hence, we get the required angle i.e. ∠AOB = 60º.

### How to Construct A 30 Degree Angle?

A 30 degree is half of a 60-degree angle. For constructing a 30-degree angle, you first construct a 60-degree angle as discussed above and bisect it. You will get two 30 degree angles.

### How to Construct A 90 Degree Angle?

We can construct a 90-degree angle either by bisecting a straight angle or using the following steps.

Step 1: Draw a line OB of any length

Step 2: Taking O as the center, and with a convenient radius, draw an arc that cuts the line segment OB at X.

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Step 3: Taking X as the center, and with the same radius, draw an arc that cuts the first arc at D.

Step 4: Taking D as the center, and with the same radius, draw an arc that cuts the first arc at C.

Step 5: Taking C and D as the center, and with the same radius, draw two arcs that cut each other at point E.

Step 6: Join OE and extend it to A.

Hence, we get the required angle i.e. ∠AOB = 90º.

### How to Construct a 45 Degree Angle?

A 45 degree is the half of 90-degree angle. For constructing a 45-degree angle, you first construct a 90-degree angle as discussed above and bisect it. You will get two 45 degree angles.

For constructing angles like 35 degrees or 22,5 degrees, you can further bisect 30-degree angle and 45-degree angle respectively.

### Facts To Remember

Euclid is a renowned Mathematician and is often known as the father of Geometry.

Euclid stated, “All right angles are congruent (i.e. equal to each other)”.

1. What are the Basic Geometric Construction Tools?

Ans: Some basic geometric construction tools are:

Ruler

Divider

Compass

Protractor

Set Squares

2. What is an Angle?

Ans: An angle is a geometric figure formed by the intersection of two rays. The common point where the two rays meet is known as vertex and the two rays are known as the arms of an angle. The angle is represented by the symbol. The angle between two line segments is the distance measured in degree or radians. Angles are important in defining and studying polygons such as triangles or quadrilaterals.

3. What are the Different Types of Angles?

Ans: There are 6 types of angles in Geometry. The names of all the 6 angles with their properties are:

**Acute Angle - **It is an angle that lies between 0º to 90º.

**Obtuse Angle -** It is an angle that lies between 90º to 180º.

**Right Angle -** It is an angle that is exactly equal to 90º.

**Straight Angle -** It is an angle that is exactly equal to 180º.

**Reflex Angle -** It is an angle that is greater than 180º and less than 360º.

**Full Rotation -** It is an angle that is exactly equal to 360º.

4. What are the Basic Geometric Constructions?

Ans: The basic geometric construction are:

Construction of line segment

Construction of line segment bisector

Construction of angle bisector

Copying of angles

Construction of Triangle

Copying a triangle

Circles of circle and tangents

Construction of polygons like square, pentagon, hexagon, etc.