Constructions of Triangles: Triangles are basic shapes that we come across in our day to day life. You are always surrounded by them. The slice of pizza, the hill nearby, the roof of your house are all triangles. Let us now study the constructions of triangles.

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## What are Triangles?

A** triangle** is a simple closed curve or polygon which is created by three line-segments. In geometry, any three points, specifically non-collinear, form a unique triangle and separately, a unique plane (known as two-dimensional Euclidean space). The basic elements of the triangle are sides, angles, and vertices. Let’s start with the constructions of triangles.

**Browse more Topics under Constructions**

- Basic Constructions
- Constructions Related to Segment and Triangles
- Construction of Tangent to a Circle

## Constructions of Triangles

### 1. Construct a scalene triangle when the base, one base angle and the sum of the lengths of the other two sides are given.

Construction:

- Draw the base BC and at the point, B makes an angle, say XBC equal to the given angle.
- Cut a line segment BD equal to AB + AC from the ray BX.
- Join DC and make an angle DCY equal to BDC.
- Let CY intersect BX at A
- ABC is the required triangle.
- In triangle ACD, ∠ACD = ∠ ADC
- So, AB = BD – AD = BD – AC
- AB + AC = BD

### 2. Construct a scalene triangle when the base, one base angle and the difference between the lengths of the other two sides are given.

**Case 1: Let AB > AC that is AB AC is given**

- Draw the base BC and at point B make an angle say XBC equal to the given angle.
- Cut the line segment BD equal to AB and AC from ray BX.
- Join DC and draw the perpendicular bisector, say PQ of DC.
- Let it intersect BX at a point A. Join AC.

**Case 2: Let AB < AC that is AC AB is given**

- Draw the base BC and at point B make an angle say XBC equal to the given angle
- Cut the line segment BD equal to AC AB from the line BX extended on opposite side of line segmentBC.
- Join DC and draw the perpendicular bisector, sayPQ of DC.
- Let PQ intersect BX at A. Join AC

### 3. Construct a scalene triangle, given its perimeter and its two base angles.

- Draw a line segment, say XY equal to BC + CA + AB.
- Make angles LYX equal to B and MYX equal to C.
- Bisect LYX and MYX. Let these bisectors intersect at a point A
- Draw perpendicular bisectors PQ of AX and RS of AY.
- Let PQ intersect XY at B and RS intersect XY at C. Join AB and AC

Then ABC is the required triangle

## Solved Examples for You

Question 1: For constructing a triangle whose perimeter and both base angles are given, the first step is to:

- Draw base of any length
- Draw base of any length = Perimeter
- To draw the base angles from the random line
- Draw base of length = 1/3 × perimeter

Solution: B is the correct option. For constructing a triangle whose perimeter and both base angles are given, the first step is to draw the base of any length = Perimeter.

Question 2: For constructing a triangle when the base, one base angle and the difference between lengths of the other two sides are given, the base length is equal to:

- The difference between the length of the other two sides.
- Given base length
- The largest side
- None of these

Solution: B is the correct option. Here the base length is equal to the given base.

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