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A triangle is a three-sided 2D shape formed by three lines known as its sides. A triangle has three vertices and three angles. The construction of triangles is one of the essential part of geometry.

Let’s understand how to construct a triangle with examples.

## How to Construct a Triangle?

A triangle has 6 basic elements – 3 sides and 3 angles. To construct any unique triangle, you should know at least 3 elements.

The following are the basic requirements to construct a unique triangle.

- All three sides are given (SSS – Side side side)
- Two sides and included angle are given (SAS – Side angle side)
- Two angles and the included side is given (ASA – Angle side angle)
- The measure of the hypotenuse and a side is given in the right triangle (RHS – Right angle hypotenuse side)

Let’s understand these constructions in detail.

### How to Construct a Triangle With SSS Property

When the length of three sides of the triangle is given, then the following are the steps to construct the required triangle.

**Step 1:** Draw a line segment $\text{AB}$, of length equal to the longest side of the triangle

**Step 2:** Now using a compass and ruler take the measure of the second side and draw an arc

**Step 3:** Again take the measure of the third side and cut the previous arc at a point $\text{C}$

**Step 4:** Now join the endpoints of the line segment to point $\text{C}$ and get the required $\triangle \text{ABC}$

### How to Construct a Triangle With SAS Property

When the length of the two sides and the angle included between them are given, then the following are the steps to construct the triangle.

**Step 1:** Draw a line segment $\text{AB}$, of length equal to the longest side of the triangle, using a ruler

**Step 2: **Put the centre of the protractor on one end of a line segment (say $\text{A}$) and measure the given angle. Join the points and construct a ray, such that the ray is nearer to the line segment $\text{AB}$

**Step 3: **Take the measure of another given side of the triangle using a compass and a ruler

**Step 4: **Place the compass at point $\text{A}$ and cut the ray at another point, $\text{C}$

**Step 5: **Join the other end of the line segment, i.e., $\text{B}$ to the point $\text{C}$

The $ \triangle \text{ABC}$ is constructed.

### How to Construct a Triangle With ASA Property

When the measures of two angles and the side included between them are given of a triangle, then it is said to be $\text{ASA}$ congruency. The following are the steps to draw a triangle with $\text{ASA}$ property.

**Step 1:** Draw a line segment $\text{AB}$, of length equal to the given side of the triangle, using a ruler

**Step 2: **At one endpoint of the line segment (say $\text{A}$) measure one of the given angles and draw a ray $\text{AR}$

**Step 3: **At another endpoint of the line segment (i.e., $\text{B}$) measure the other angle using a protractor and draw the ray $\text{BQ}$, such that it cuts the previous ray at a point $\text{P}$

**Step 4: **Join the previous point $\text{P}$, with both the endpoints $\text{A}$ and $\text{B}$ of the line segment $\text{AB}$, to get the required triangle

### How to Construct a Triangle With RHS Property

When the hypotenuse side and any one of the other two sides of the right triangle are given, then it is $\text{RHS}$ property. The following are the steps to draw a triangle with $\text{RHS}$ property.

**Step 1:** Draw the line segment $\text{AB}$, equal to the measure of the other side

**Step 2: **At one endpoint, say $\text{AB}$, of line-segment measure the angle equal to $90^{\circ}$ degrees and draw a ray, $\text{AR}$

**Step 3: **Measure the length of the hypotenuse and draw an arc to cut the ray $\text{AR}$ at a point $\text{P}$

**Step 4: **Join the points $\text{P}$ and $\text{B}$ to get the required right triangle

## How to Construct a Triangle – Special Cases

After understanding the steps for the construction of general triangles, let’s now understand how to construct special triangles when

- sum of the two sides is given
- difference between the two sides is given

### How to Construct a Triangle With Base Angle and the Sum of the Other Two Sides

Let’s now understand the procedure of construction of a triangle where the base of a triangle, its base angle, and the sum of the other two sides are given.

For constructing a $\triangle \text{ABC}$ such that base $\text{BC}$, base angle $\angle \text{B}$ and the sum of the other two sides, i.e. $\text{AB} + \text{AC}$ are given, the following steps of construction are used.

**Step 1:** Draw the base $\text{BC}$ of $\triangle \text{ABC}$ as given and construct $\angle \text{XBC}$ of the given measure at $\text{B}$

**Step 2:** Keeping the compass at point $\text{B}$ cut an arc from the ray $\text{BX}$ such that its length equals $\text{AB} + \text{AC}$ at point $\text{P}$ and join it to $\text{C}$

**Step 3:** Now measure $\angle \text{BPC}$ and from $\text{C}$, draw an angle equal to $\angle \text{BPC}$

### How to Construct a Triangle With Base Angle and the Difference Between the Other Two Sides

Let’s now understand the procedure of construction of a triangle where the base of a triangle, its base angle, and the difference between the other two sides are given.

For constructing $\triangle \text{ABC}$ such that base $\text{BC}$, base angle $\angle \text{B}$ and difference of the other two sides, i.e. $\text{AB} – \text{AC}$ or $\text{AC} – \text{AB}$ is given, then for constructing triangles such as these two cases can arise

- $\text{AB} \gt \text{AC}$
- $\text{AC} \gt \text{AB}$

The following steps of construction are followed for the two cases:

#### Steps of Construction When $\text{AB} \gt \text{AC}$

**Step 1: **Draw the base $\text{BC}$ of $\triangle \text{ABC}$ as given and construct ∠XBC of the required measure at B as shown.

**Step 2:** From the ray, BX cut an arc equal to AB – AC at point P and join it to C as shown

**Step 3:** Draw the perpendicular bisector of PC and let it intersect BX at point A as shown:

**Step 4:** Join AC, ∆ABC is the required triangle.

#### Steps of Construction When $\text{AC} \gt \text{AB}$

**Step 1:** Draw the base BC of ∆ABC as given and construct ∠XBC of the required measure at B as shown.

**Step 2:** On the ray BX cut an arc equal to AB – AC at point P and join it to C. In this case, P will lie on the opposite side to the ray BX. Draw the perpendicular bisector of PC and let it intersect BX at point A as shown

**Step 3:** Join the points A and C, and hence ∆ABC is the required triangle.

## Practice Problems

- Construct a $\triangle \text{PQR}$ in which $\text{PQ } = 5.4 \text{cm}$, $\angle \text{Q} = 60^{\circ}$ and $\text{PR} – \text{PQ} = 2.3 \text{cm}$.
- Construct a $\triangle \text{XYZ}$ in which $\angle \text{Y} = 45^{\circ}$, $\angle \text{Z} = 75^{\circ}$ and $\text{XY} + \text{YZ} + \text{ZX} = 12 \text{cm}$.
- Construct a right-angled triangle whose be is $3.8 \text{cm}$ and hypotenuse is $5.6 \text{cm}$.
- Construct a $\triangle \text{ABC}$ in which $\angle \text{B} = 60^{\circ}$, $\angle \text{C} = 30^{\circ}$ and the length of the perpendicular from the vertex $\text{A}$ is $5.3 \text{cm}$.

## FAQs

### What is a triangle construction?

Drawing a triangle with specific dimensions considering all the related properties of triangles is called triangle construction. Construction of any type of triangle can be done with the help of a ruler, a compass, or a protractor.

### Which criterion is used for the construction of a triangle?

To construct a triangle, the following criteria are used.

a) **SSS criterion:** A triangle in which all three sides are known.

b) **ASA criterion:** A triangle in which two angles and one side are known.

c) **SAS criterion:** A triangle in which two sides and one angle are known.

d) **RHS criterion:** A triangle in which the hypotenuse and another side are known.

### How do you construct a triangle with 3 sides?

Draw a line with the length of the longest side. Draw two arcs from the two endpoints of the line drawn such that they intersect each other. Join the intersecting point with the vertices of the longest side.

## Conclusion

Drawing a triangle with specific dimensions considering all the related properties of triangles is called triangle construction. You can construct a unique triangle using a ruler, a compass, or a protractor. The basic criteria used to construct triangles are SSS, SAS, ASA, and RHS criteria.

## Recommended Reading

- Median of a Triangle(Definition & Properties)
- Types of Triangles – Definition & Examples
- What is Triangle in Geometry – Definition, Shapes & Examples
- Pair of Angles – Definition, Diagrams, Types, and Examples
- Construction of Angles(Using Protractor & Compass)
- Types of Angles in Maths(Acute, Right, Obtuse, Straight & Reflex)
- What is an Angle in Geometry – Definition, Properties & Measurement
- How to Construct a Tangent to a Circle(With Steps & Pictures)
- Tangent of a Circle – Meaning, Properties, Examples
- Angles in a Circle – Meaning, Properties & Examples
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- How to Draw a Circle(With Steps & Pictures)
- What is a Circle – Parts, Properties & Examples
- How to Construct a Perpendicular Line (With Steps & Examples)
- How to Construct Parallel Lines(With Steps & Examples)
- How To Construct a Line Segment(With Steps & Examples)
- What are Collinear Points in Geometry – Definition, Properties & Examples
- What is a Transversal Line in Geometry – Definition, Properties & Examples
- What are Parallel Lines in Geometry – Definition, Properties & Examples
- What is Concurrent lines in Geometry – Definition, Conditions & Examples
- What is Half Line in Geometry – Definition, Properties & Examples
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- Difference Between Axiom, Postulate and Theorem
- Lines in Geometry(Definition, Types & Examples)
- What Are 2D Shapes – Names, Definitions & Properties
- 3D Shapes – Definition, Properties & Types