How to Construct a Triangle(With Steps, Diagrams & Examples)

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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

1. Construct a $\triangle \text{PQR}$ in which $\text{PQ } = 5.4 \text{cm}$, $\angle \text{Q} = 60^{\circ}$ and $\text{PR} – \text{PQ} = 2.3 \text{cm}$.
2. Construct a $\triangle \text{XYZ}$ in which $\angle \text{Y} = 45^{\circ}$, $\angle \text{Z} = 75^{\circ}$ and $\text{XY} + \text{YZ} + \text{ZX} = 12 \text{cm}$.
3. Construct a right-angled triangle whose be is $3.8 \text{cm}$ and hypotenuse is $5.6 \text{cm}$.
4. 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

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.

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