CONSTRUCTING TRIANGLES

Triangle is a polygon that has three sides. We can construct it if we know its two angles, as we can find its third angle by using the property that the sum of all interior angles of a triangle is 180 degrees and if we know its sides.

Types of Triangles

Basically, triangles are of three types:-

1. Equilateral Triangle – The triangles which have a measure of three sides and three angles equal are known as an equilateral triangles.
2. Isosceles Triangle – The triangles which have a measure of their two sides equal to their two angles are known as an isosceles triangle.
3. Scalene Triangle – The triangles which have the measure of three sides different along with angles are not the same as the triangle, known as the Scalene triangle.

Important Properties to Construct a triangle

Before starting doing construction of a triangle, you should know about the following properties:-

1. SSS – When all three sides of a triangle are given, then there will apply SSS i.e., SIDE SIDE SIDE property
2. SAS – This property consists of two sides and one angle of the triangle known as the SIDE ANGLE SIDE property
3. ASA –  Under this property, we have two angles and one side of a triangle, it is known as the ANGLE SIDE ANGLE property
4. RHS – This property is applicable in Right angle triangles, according to this we have a measure of the hypotenuse and a side of the triangle.

These all are Congruence Properties and are important for constructing a triangle.

Construction using SSS property

When you know the measure of three sides of the triangle, construct the triangle using the following steps:-

1. Draw the longest line segment and name it AB.
2. Then, using a ruler and compass, make an arc that should be equal in length to another side
3. Now, cut that arc by making an arc of the length of the third side. This will be point C.
4. Meet the ends of line segment AB to the intersecting point of the arc.
5. Hence, We have constructed Triangle ABC.

Construction using SAS property-

When you know the measure of two sides and one angle of the triangle, construct the triangle using the following steps:-

Step – 1: Draw the line segment AB using a ruler with the length of the longest side of the triangle.

Step – 2: Now, draw the given angle at the end of the line segment by putting the center of the protractor at the end of a line segment, and make a point and join this point with AB.

Step – 3: Using a Compass and ruler, draw the arc of length equal to the measure of the other side of the triangle. Draw this arc on the line of the angle.

Step – 4: Join the point at which the arc cuts the other line (This will be point C) to the endpoint of line segment AB.

Step – 5: Hence, We have constructed Triangle ABC.

Construction using ASA property-

When you know the measure of two angles and one side of the triangle, construct the triangle using the following steps:-

1. Using Ruler, draw the line segment AB equals to a given length.
2. At one endpoint of AB, draw the given angle using a protractor and joint the point by a ray, say AR.
3. At another endpoint of AB, draw another angle and draw the arc by a new ray say BQ which cuts the previous arc at point C.
4. Now, join point C with the endpoints of line segment AB.
5. Hence, We have constructed Triangle ABC.

Construction using RHS property-

When you know the measure of the hypotenuse and one side of the triangle, construct the triangle using the following steps:-

1. Using a ruler, draw the line segment AB of the length of the hypotenuse.
2. At one endpoint of line segment AB, draw the right angle, 90 degrees with the help of a ray AR.
3. Draw the other side using a compass and ruler, make the arc which cuts the previous arc, say point C.
4. join the line segments of AB with point C.
5. Hence, We have constructed Triangle ABC.-

Construction of a 30-60-90 degrees triangle

1. Draw the line segment AB which will be the hypotenuse of the triangle, As you know in a right-angle triangle, the hypotenuse is opposite to a 90-degree angle, so at the ends of AB, angles will be 30 and  60 degrees.
2. At one endpoint of AB, say A, draw the angle of 30 degrees using a compass and draw an arc with ray AR.
3. At another endpoint of AB, say B, draw the angle of 60 degrees using a compass and draw another arc that cuts the previous arc, say this point C.
4. Join the Line segment with point C.
5. Hence, We have constructed Triangle ABC.

   Conclusion-

   We have studied the construction of a triangle with the help of a ruler, compass, and protractor. And also about the properties of congruence of a triangle, which is an important part of the construction of a triangle.