We generally construct a triangle based on the congruency conditions of the triangle. A triangle is a polygon made up of three line segments. Thus, a triangle has three sides and three angles. Using geometrical tools like a protractor, compass, ruler and scale, we can construct the triangle. However, for the Construction of Triangles, we need to know about the properties of the triangle.

Three line segments form a triangle, which is a simple closed curve or polygon. Any three non-collinear points create a unique triangle and, independently, a unique plane in geometry (known as two-dimensional Euclidean space). The triangle’s basic constituents are sides, angles, and vertices. In this article, we will understand how to construct different types of triangles. Continue reading to know more.

Triangle and its Types

A triangle is a three-dimensional polygon, which has only three sides, three angles and three vertices. There are various types of triangles based on sides, angles, and other factors.

Triangles Based on Sides

1. Equilateral Triangle: A triangle with three equal sides and all three equal angles is known as an equilateral triangle.

2. Isosceles Triangle: A triangle with any two equal sides is known as the Isosceles triangle.

3. Scalene Triangle: A triangle with all sides unequal is known as a scalene triangle.

Triangles Based on Angles

1. Acute-angled Triangle: A triangle in which all angles less than \({90^ \circ }.\)

2. Right-angled Triangle: A triangle with an angle \({90^ \circ }\), It is called a right-angled triangle.

3. Obtuse-angled Triangle: A triangle in which one angle more than \({90^ \circ },\) It is called an obtuse-angled triangle.

Rules and Properties of Triangles

1. A triangle with vertices \(P,\,Q,\,R\) is denoted as \(\Delta PQR.\)
2. By angle sum property, the sum of all angles in a triangle is \({180^ \circ }.\)
3. In the triangle, the sum of any two sides is always greater than the third side.
4. The difference between any two sides of the triangle is always less than the third side.
5. An exterior angle of the triangle is the sum of opposite interior angles.
6. The side opposite to the largest side of the angle is greater in a triangle.

Construction of Triangles – SSS

Construction of a triangle when the measurements of three sides are given is explained by using the example:

Example: Construct a triangle with sides \(3{\mkern 1mu} \,{\rm{cm}},\,{\rm{4}}\,{\rm{cm}},\,{\rm{and}}\,{\rm{5}}\,{\rm{cm}}{\rm{.}}\)

1. Step –\({\rm{1}}\) : Draw a line of length \(BC{\mkern 1mu} = {\mkern 1mu} 5\,{\rm{cm}}.\)

2. Step –\({\rm{2}}\): Draw an arc by using a compass from point \(B\) such that \(BA\, = \,4\,{\rm{cm}}.\)

3. Step-\({\rm{3}}\): Draw another arc from point \(C\) with a length of \(CA\, = \,3\,{\rm{cm}},\) by using a compass.

4. Step-\({\rm{4}}\) Join the intersection point \((A)\) with other points \({\rm{(}}B,\,C{\rm{)}}\) of the line.

\(\Delta ABC\) is the required triangle.

Construction of Triangles – ASA

Construction of triangle, when two angles and one side are given, is explained by using an example:

Example: Construct a triangle with side \(AB\, = \,4\,{\rm{cm}}\) and \(\angle A\, = \,{30^ \circ },\angle B = {60^ \circ }\)

1. Step–\({\rm{1}}\): Draw a line of length \(AB\, = \,4\,{\rm{cm}}\) by using scale and pencil.

2. Step-\({\rm{2}}\): Now, draw an angle \({30^ \circ }\) from point \(A\) by using the protractor.

3. Step-\({\rm{3}}\): Draw an angle \({\rm{6}}{{\rm{0}}^ \circ }\) From point \(B\) by using the protractor.

4. Step-\({\rm{4}}\): Mark the intersection point as \(C,\) Thus,\(\Delta ABC\) is the required triangle.

Construction of Triangles – SAS

Let us discuss the construction of a triangle when the measurements of two sides and one angle is given.

Example: Construct a triangle with sides \(AB = \,3\,{\rm{cm}},\,AC\, = 4\,{\rm{cm}}\) and \(\angle A\, = \,{30^ \circ }.\)

1. Step–\({\rm{1}}\): Draw a line \(AB\) of length \(3\,{\rm{cm}}\) by using the scale and pencil.

2. Step–\({\rm{2}}\): Draw an angle of \({30^ \circ }\) from point \(A\) by using a protractor.

3. Step-\({\rm{3}}\): Draw an arc of length \(4\,{\rm{cm}}\) from point \(A\) by using a compass, which cuts the previous line at \(C.\)

4. Step-\({\rm{4}}\): Join \(BC.\) Thus, \(\Delta ABC\) is the required triangle.

Construction of Triangles – RHS

Let us discuss the construction of the triangle when the measurements of the side and hypotenuse of the right-angled triangle are given.

Example: Construct a triangle with sides \(AB = 3\,{\rm{cm}},BC = 5\,{\rm{cm}}\,{\rm{and}}\,\angle A = {\rm{ }}90^\circ {\rm{ }}.\)

1. Step–\({\rm{1}}\): Draw a line \(AB\) of length \(3\,{\rm{cm}}\) by using the scale and pencil.

2. Step–\({\rm{2}}\): Draw an angle of \({90^ \circ }\) from point \(A\) by using a protractor.

3. Step-\({\rm{3}}\): Draw an arc of length \(5\,{\rm{cm}}\) from point \(B\) by using a compass, which cuts the previous line at \(C.\)

4. Step–\({\rm{4}}\): Join \(BC.\) Thus, \(\Delta ABC\) is the required triangle.

Construction of Equilateral Triangles

We know that a triangle with measures of all three sides is equal is known as an equilateral triangle.

Example: Construct an equilateral triangle with sides \(8\,{\rm{cm}}.\)

1. Step-\({\rm{1}}\): Draw a line segment of a given length \((8\,{\rm{cm)}}\) using scale and pencil.

2. Step-\({\rm{2}}\): Place the point of the compass at one end of the line segment and draw an arc of the given length \({8\,{\rm{cm}}}\) by using a compass.

3. Step-\({\rm{3}}\): Place the point of the compass at the other end of the line and draw an arc of the same length by using a compass.

4. Step-\({\rm{4}}\): Mark the intersection and join with other points of the line. The figure gives the required equilateral triangle with a side \(8\,{\rm{cm}}.\)

Construction of Isosceles Triangles

A triangle in which any two sides are the same, and one unequal side is known as an isosceles triangle.

Example: Construct an isosceles triangle with sides \({\rm{7}}{\mkern 1mu} {\rm{cm}},{\mkern 1mu} {\rm{7}}\,{\rm{cm}}{\mkern 1mu} \,{\rm{and}}\,{\rm{4}}{\mkern 1mu} {\rm{cm}}\)

1. Step-\({\rm{1}}\): Draw a line of length \({\rm{4}}\,{\rm{cm}}\) by using scale and pencil.

2. Step-\({\rm{2}}\): Place the pointer of the compass at one end of the line and draw an arc of \({\rm{7}}\,{\rm{cm,}}\) by using the compass.

3. Step-\({\rm{3}}\): Place the pointer of the compass at another end of the line and draw an arc \({\rm{7}}\,{\rm{cm,}}\) by using the compass.

4. Step-\({\rm{4}}\): Join the intersection point of arcs to other endpoints of the line. The figure gives the required isosceles triangle with sides \({{\rm{7}}{\mkern 1mu} {\rm{cm}},{\mkern 1mu} {\rm{7}}{\mkern 1mu} {\rm{cm}}{\mkern 1mu} \,{\rm{and}}\,{\rm{4}}{\mkern 1mu} {\rm{cm}}}\)

Construction of Triangles: Different Cases

Case 1: Construction of triangle, when the base, one base angle and the sum of lengths of other two sides:

Example: Construct a triangle \({\rm{ABC,}}\) in which \(BC{\mkern 1mu} = {\mkern 1mu} 7{\mkern 1mu} {\rm{cm}},{\mkern 1mu} \,\angle B{\mkern 1mu} = {\mkern 1mu} {75^^\circ }\) and \(AB\, + \,AC\, = \,13\,{\rm{cm}}{\rm{.}}\)

Steps of construction:

1. Draw a line segment \(BC\, = \,7\,{\rm{cm}}\)
2. At point \(B\), draw an angle \(\angle CBX\, = \,{75^ \circ }\) using the protractor.
3. From point \(B\) cut the previous line with an arc of length \(13\,{\rm{cm}}\)
4. Join \(DC\)
5. Draw the perpendicular bisector \(LM\) of \(CD,\) which intersects \(BD\) at \(A.\)
6. Join \(AC.\) Thus, the \(ABC\) is the required triangle.

Case 2: Construction of triangle, when the base, one base angle and difference of lengths of other two sides:

Example: Construct a triangle \(ABC\) in which \(BC = \,8\,{\rm{cm, }}\angle B\, = \,{45^ \circ }\) and \(AB\, – \,AC\, = \,3.5\,\,{\rm{cm}}.\)

Steps of construction:

1. Draw a line segment \(BC = \,8\,{\rm{cm}}.\)
2. At point \(B\), draw an angle \(\angle CBX\, = \,{45^ \circ }\) by using the compass.
3. From point \(B\), cut the previous line with an arc of length \(3.5\,{\rm{cm}}.\)
4. Join \(DC.\)
5. Draw the perpendicular bisector \(LM\,\,of\,CD\) which intersects \(BX\) at \(A.\)
6. Join \(AC.\) Thus, the \(ABC\) is the required triangle.

Case 3: Construction of triangle, when two base angles and the sum of lengths of all three sides (perimeter):

Example: Construct a triangle \(XYZ,\) in which \(\angle X = {30^o},\,\angle Z = {90^o}\) and \(XY + YZ + ZX = 11\;{\rm{cm}}.\)

Steps of Construction:

1. Draw a line segment \(AB = 11\,{\rm{cm}}\) by using scale and pencil.
2. At a point \(A\), draw an angle \(\angle BAP\, = \,{30^ \circ }\) using a compass.
3. At a point, \(B\) draw an angle \(\angle ABR\, = \,{90^ \circ }\) using the compass.
4. Draw the bisectors of \(\angle BAP\) and \(\angle ABR\), which intersect each other at \(Y\)
5. Join \(AY\) and \(BY\)
6. Draw the perpendicular bisectors \(LM\) and \(ST\) of \(AY\) and \(BY\) respectively.
7. \(LM\) and \(ST\) intersect \(AB\) at \(X\) and \(Z\), respectively.
8. Join \(XY\) and \(YZ.\)
9. Thus, the triangle \(XYZ\) is the required triangle.

Solved Examples: Construction of Triangles

Q.1. Construct a triangle \(PQR\), in which \(PQ\, = \,5\,\,{\rm{units,}}\), \(QR\, = \,6\,\,{\rm{units}}\) and \(PR\, = \,3.5\,\,{\rm{units}}.\)
Ans: The steps are as follows:
Step 1: Draw a line segment \(QR\) measuring \(6\,{\rm{units}}.\)
Step 2: At point \(Q\) take a measure of \(5\,{\rm{units}}\) in the compass and draw an arc
Step 3: At point \(R\) take a measure of \(3.5\,{\rm{units}}\) in the compass and draw an arc intersecting the previous arc.
Step 4: Join the lines \(PQ\) and \(PR\) Thus, the triangle \(PQR\) is the required one.

Q.2. Construct triangle \(XYZ\) , in which \(XY\, = \,7\,{\rm{units,}}\,\angle X = {40^ \circ },\,\angle Y = \,{105^ \circ }.\)
Ans: Following are the steps:
Step 1: Draw a line segment \(XY\, = \,7\,{\rm{units}}\) by using scale and pencil.
Step 2: At point \(X\) drawn an angle \(\angle X = {40^ \circ }\) with the protractor.
Step 3: At point \(Y\), drawn an angle \(\angle Y = {105^ \circ }\) with the protractor.
Step-4: Mark the intersection point as \(Z\).
Step 5: Join the points \(XZ\), \(YZ\).
Step-6: Thus, the triangle \(XYZ\) is the required one.

Q.3. Draw a right-angled isosceles triangle where one of the bases is \(4\,{\rm{units}}\)
Ans: The steps to follow are listed below:
Step-1: Draw a line \(QR\, = \,4\,{\rm{units}}\) with scale and pencil.
Step-2: At point \(Q\) draw an angle of \({\rm{9}}{{\rm{0}}^ \circ }\) with the protractor.
Step-3: At point \(Q\) draw an arc of length \(QR\, = \,4\,{\rm{units}}\) by compass.
Step-4: Join \(PR\). Thus, triangle \(PQR\) is the required one.

Q.4. Construct an equilateral triangle \(XYZ\) with sides \(6\) units.
Ans: All the steps to construct are listed below:
Step-1: Draw a line of length \(YZ = 6\,{\rm{units}}\) by using scale and pencil.
Step-2:Place a point of the compass at point \(Y\), and draw an arc of length \(6\,{\rm{units}}\) by using a compass.
Step-3: Place a point of the compass at point \(Z\), and draw an arc of length \(6\,{\rm{units}}\) by using a compass.
Step-4: Mark the intersection as \(X\), andjoin \(XY\), \(XZ\)

Q.5. Draw an isosceles triangle of sides \(6\,\,{\rm{units}},\,6\,\,{\rm{units, 5}}\,{\rm{units}}\)
Ans: These are the steps to follow:
Step 1: Draw a line segment \(BC\, = \,5\,{\rm{units}}\)
Step 2: At point \(B\) draw an arc with a measure of \(6\,{\rm{units}}{\rm{.}}\)
Step 3: Now, at point \(C\) draw an arc with the same measure of \(6\,{\rm{units}}{\rm{.}}\)
Step-4: Mark the intersection point as \(A\)
Step-5: Join \(AB,AC\) to form an isosceles triangle \(ABC\).

Summary

The construction of a triangle is very useful in various parts of Mathematics, such as practical geometry. In this article, we have learnt about the construction of triangles based on the different criteria given.

Here, we also have learnt about the construction of special types of triangles such as equilateral, isosceles triangles. This article would help us to know the various properties and constructions of the different triangles.

Frequently Asked Questions (FAQ): Construction of Triangles

Q.1. What are the uses of the construction of triangles?
Ans: The uses of construction of triangles are:
1. In the construction of towers, bridges.
2. In the application of the Pythagoras theorem.

Q.2. What is triangle construction?
Ans: Drawing a triangle with some specific conditions by keeping the view of properties of the triangle is called triangle construction.

Q.3. What are the criteria used for the construction of triangles?
Ans: The criterion used for the construction of triangles are
1. SSS – When the measurements of all three sides are known.
2. SAS – When measurements of two sides and an angle is known.
3. ASA – When measurement of two angles and the side known.
4. RHS – When the measurement of base and hypotenuse known.

Q.4. List the geometrical tools used in the construction of a triangle.
Ans: The geometrical tools used in the construction of triangles are scale, compass, protractor, ruler, etc.

Q.5. How do you construct a triangle with three sides?
Ans: 1. Draw a line with the length of the longest side.
2. Draw arcs from the two endpoints of the line.
3. Join the intersecting point with other points of the line.