Introduction to Constructions

Look around yourself. What do you see? You will find different objects around you. Are all these objects the same? Do they have the same shapes? No, not every object is of the same shape. You are quite familiar with different geometric shapes. In this section, we will talk about constructions of the geometric shapes. Let us start to learn more about geometric constructions.

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Introduction to Geometric Constructions

As you are familiar with various shapes, you can draw them with your hands. You are well aware with the geometric constructions of a line segment of a certain measurement, a square, a rectangle or a triangle with the help of a ruler. In this section, we are going to learn some more geometric constructions with the help of a compass, a ruler, and a protector.

Let us learn how to draw a few angles, two parallel lines, and some triangles.

Construction of a Line Parallel to a Given Line

Have you ever seen railway tracks? What is so unique about them? They are always at a fixed distance apart from one another. The edges of a ruler are also an example of parallel lines. Let us learn the steps of the geometric constructions of the parallel lines. There are many ways in which one can construct parallel lines.

Construction of Parallel Lines through a Point not on the Line

Take out your compass and a ruler and let’s start the geometric constructions.

• Draw a line l of any length of your wish.
• Draw two points one on the line and the other outside the line. Name them A and B respectively.
• From the point, A on the line l draw a dotted line to join the point B.
• Taking A as the center and with any desired radius draw an arc cutting the line l at C. The same arc cuts the line AB at D.
• Now with the same chosen radius, draw an arc with B as a center. Name this arc PR cutting AB at Q.
• Adjust your compass such that the distance between the tip of the pencil and the tip of the needle of the compass is the distance between AC.
• With the same adjustment, place the tip of the compass on B and cut the arc PR at O.
• Draw a straight line passing through the points O and B. This line m will be parallel to the line l.

Construction of Triangles

One can find various forms of triangles. Some of them will have all the three sides of the same length. There are other triangles in which only two sides are equal or none of them are of equal length. Also, we can find some triangles which are different from one another on the basis of the measurement of their vertex angles. The triangles may be an acute-angled triangle or right-angled or obtuse-angled based on the angle of measurement.

Construction of a Triangle when its Three Sides are known

Here, we will learn the steps for the geometrical construction of a triangle with three known sides. Suppose we have to construct a triangle with sides of 8cm, 3 cm, and 5 cm.

• The first step is to draw a rough sketch always.
• Draw a line segment AB of length 8 cm.
• From A as the point of center, we make an arc with a compass at a distance of 3 cm. You can also take the distance of 5 cm if you wish.
• Taking B as a center, draw an arc with a compass at a distance of 5 cm. This arc will cut the previously made arc. Name this point of intersection of an arc as C.
• Join A to C and B to C. We have our triangle.

This is the SSS criterion.

Construction of a Triangle when its Two Sides and an Angle between them are known

Suppose we have to construct a triangle with sides of 8cm and 5 cm, and an angle of 60° between them.

• The first step is to draw a rough sketch always.
• Draw a line segment AB of length 8 cm.
• From A as the point of center, we make an angle of 60° with a compass.
• Draw a line from this point A passing through this arc of 60°.
• Set the compass at a distance of 5 cm. From the point, A cut the line by making an arc with the help of the compass. Name this point C.
• Join B to C. We have our triangle.

This is the SAS criterion.

Construction of a Triangle when its Two Angles and a Length of the Side between them are known

Suppose we have to construct a triangle with angles of 60° and 100° and a side of 8 cm between them.

• The first step is always to draw a rough sketch.
• Draw a line segment AB of length 8 cm.
• From A as the point of center, we make an angle of 60° with a compass.
• From B as the point of center, we make an angle of 100° with the help of a protector.
• Extend the line from the point A and B until they intersect each other.
• Name this point of intersection as C.
• Join A to C and B to C. We have our triangle.

This is the ASA criterion.

Construction of a Right Angled Triangle with Two Sides given

Suppose we have to construct a right-angled triangle. The lengths of the two sides are 5 cm and 4 cm. Let us assume that the length of the hypotenuse of a right-angled triangle is 5 cm. The length of the base is 4 cm. It is clear that the length of the perpendicular to the base is of 3 cm (by the Pythagoras Theorem).

• The first step is to draw a rough sketch.
• Draw a line segment AB of length 4cm.
• Draw a perpendicular bisector at A. Name it AO.
• Take the point B as the center. Draw an arc of length 5 cm with the help of a compass. This arc will intersect the perpendicular bisector AO. Name the point of intersection as C.
• Join BC.

The triangle ABC is a triangle right angled at A i.e., ∠CAB = 90°.

This is the RHS criterion.

Solved Example for You

Question 1: Suppose the two sides of the triangle are of the same length of 5.5 cm. One of the angles made by this side of the triangle is 80°. Find the other two angles. Construct the triangle.

Answer : Since the two sides of the triangle are of the same length, the angles made by them with the base will always be the same. The angles made by them are of 80° each. By the property of the sum of the interior angles of a triangle, we have

x + 80° + 80° = 180° (let the third angle is of x degrees).

or, x = 180° – 160° = 20°.

The required triangle

Question 2: What is a perpendicular bisector of a triangle?

Answer: The perpendicular bisector of a side of a triangle refers to a line perpendicular to the side and one which passes through its midpoint. The three perpendicular bisectors of the sides of a triangle meet in a single point which we refer to as the circumcenter. It is halfway from the vertices of the triangle.

Question 3: How is geometry used in real life?

Answer: Geometry has a lot of practical uses on a daily basis. For instance, when you need to measure the circumference, area and volume. Moreover, also when there is a need for building or creating something. Further, geometric shapes have a significant role to play in our lives that extend to recreational activities like sports, food design, video games and more.

Question 4: What is the difference between straightedge and ruler?

Answer: The difference is that we use a straightedge merely to make or draw a straight line but not for measuring the length because it doesn’t consist of measurement gradients. On the other hand, a ruler consists of two parallel lines or edges that have centimetres and millimetres on one side and has got inches on the other side.

Question 5: Who invented geometric construction?

Answer: Geometry is said to be one of the two fields of pre-modern math. The other one is arithmetic which means the study of numbers. You see that classic geometry was centred on compass and straightedge constructions. Then, Euclid revolutionized it by introducing rigour plus the axiomatic method that is even relevant today.