 # Constructing Triangles

A triangle is a three-sided polygon. It has three sides, three vertices  and three angles. We know that a unique triangle can be constructed if
(i) all three sides are given
(ii) two sides and included angle is given
(iii) two angles and the included side is given
(iv) the measure of hypotenuse and a side is given in a right triangle.

For constructing triangles from given data we generally make use of the given congruency conditions and construct the required triangle. Let us discuss the technique of constructing triangles when specific conditions are given:

Constructing Triangles 1:

Given base of a triangle, its base angle and sum of other two sides

For constructing ∆ABC such that base BC, base angle∠B and sum of other two sides i.e. AB + AC is given the following steps of construction is followed:

Steps of Construction of a Triangle:

1. Draw the base BC of ∆ABC as given and construct ∠XBC of the required measure at B as shown. 2. Keeping the compass at point B cut an arc from the ray BX such that its length equals to AB + AC at point P and join it to C as shown 3. Now measure ∠BPC and from C draw an angle equal to ∠BPC as shown ∆ABC is the required triangle. This can be proved as follows:

 Sl.No Statement Reason Base BC and ∠B are drawn as given. Now in ∆ACP, 1 ∠ACP = ∠APC By Construction 2 AC = AP ∆ACP is isosceles 3 AB = BP – AP = BP – AC From Statement 2 4 AB + AC = BP Proved

Constructing Triangles 2:

Given base of a triangle, its base angle and difference of other two sides

For constructing ∆ABC such that base BC, base angle∠B and difference of other two sides i.e. AB – AC or  AC-AB is given, then for constructing triangles such as these two cases can arise:

1. AB > AC
2. AC > AB

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

Steps of Construction if AB > AC:

1. Draw the base BC of ∆ABC as given and construct ∠XBC of the required measure at B as shown. 2. From the ray BX cut an arc equal to AB – AC at point P and join it to C as shown

3. Draw the perpendicular bisector of PC and let it intersect BX at point A as shown: 4. Join AC, ∆ABC is the required triangle. Steps of Construction if AC > AB:

1. Draw the base BC of ∆ABC as given and construct ∠XBC of the required measure at B as shown. 2. From 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 3. Join point A and C, ∆ABC is the required triangle.