Construction of Triangles

Concept of Triangles

We all are already aware of the concept of Triangle. A Triangle may be a three-sided Polygon made from Three sides having three angles. It is to be noted that within the construction of a triangle the three sides and angle may or might not be equal in dimensions.

Practical geometry, which deals with the development of various geometrical figures, is a crucial branch of Geometry. Using a set of geometrical tools such as rulers, compasses and protractors, different shapes like squares, triangles, circles, hexagons, etc. can be constructed. The only condition is that you should be aware of the properties of these figures that set them apart from one another. You are already aware of the construction of lines, angles, bisectors, etc. This knowledge is extended for triangle construction. But before we move to that, in this section, we bring to you all the properties of triangles that you need to keep in mind before moving over to triangle construction.

Example 1:-

Construct a triangle when the bottom , one base angle and  the sum of the lengths of the opposite two sides are given

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Draw the bottom BC and at the point B , B makes an angle, say XBC, adequate to the given angle.

Cut a line segment BD adequate to AB + AC from the ray BX.

Join DC and make an angle DCY adequate to BDC.

Let CY intersect BX at A

ABC is the required triangle.

In triangle ACD, ∠ACD = ∠ ADC

So, we can write  AB = BD – AD = BD – AC

AB + AC = BD

Example 2:-

Now for constructing a triangle whose perimeter and both base angles are given, the first step is to:

1. Draw base of any length

2. Draw base of any length  = Perimeter

3. To draw the base angles from the given or random line

4. Draw the base with length = 1/3 × perimeter

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

Draw Different Type of Triangles -

1. Construction of a Congruent Triangle :-

We're working on a neighborhood with four different models, and this is our first one. We want to copy it, which means making a congruent triangle. Thank goodness we live in a two-dimensional geometry world, or we'd probably need to know things about plumbing and electrical work.

Let's start with a point. Let's call it D. Now, draw a ray using the ruler from D to form our base, or foundation of the house (see video starting at 01:00 to see these actions). Next, take the compass and measure the distance from A to C. Use this width to draw an arc that hits our ray. Where it hits is point F, the equivalent of point C on the model.

Next, use the compass to measure the distance from A to B. Again, use this to draw an arc around where the top of the new triangle should be. Don't add a point yet. We're not sure exactly where the top of our house will be.

Let's measure C to B and draw another arc, this time from point F. Where these arcs meet is our final point, point E. Now, just connect D to E and F to E, and we have a congruent triangle.

2. SSS Construction of Triangles

Construct ΔXYZ during which XY = 4.5 cm, YZ = 5 cm and ZX = 6cm. 

Step 1. Draw a line YZ of length 5 cm.

Step 2. From point Y, the point X is at a distance of 4.5 cm. So, with Y as the center of the line, draw an arc of radius 4.5 cm.

Step 3. From point Z, point X is at a distance equal to 6 cm. So, with Z as the center of the line, draw an arc of radius 6 cm. 

Step 4. X has got to get on both the arcs drawn. So, it is the point of intersection of arcs. Mark the point of interaction of arcs as X. Join XY and XZ. ΔXYZ is the required triangle.

3. Construction of an isosceles triangle

A triangle may be a triangle with two equal side lengths and two equal angles. Sometimes you'll have to draw a triangle given limited information. If you know the side lengths, base, and altitude, it is possible to do this with just a ruler and compass (or just a compass, if you are given line segments). Using a protractor, you'll use information about angles to draw a triangle .

4. Case where AB > AC or AC > AB

We have been given the base of a triangle, the base angle of the triangle and difference of the other two sides of the triangle.

Now for constructing the triangle (∆ABC) such that base BC,the base angle∠B and difference of the other two sides (that is AB – AC or  AC-AB )is given to us, then for constructing such triangles these two cases can arise:

1. Length of AB > Length of AC

2. Length of AC > Length of AB

The Following Steps of Construction are Followed for the Given Two Cases -

Here are the Steps of Construction if AB > AC:

1. First draw the base BC of  the triangle ∆ABC as given and construct XBC of the required measure at B as shown below .

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2. Now from the ray BX ,cut an arc that is equal to AB – AC at point P and join it to C as shown

3. Now , draw the perpendicular bisector of PC and let it intersect at the point BX at point A .   

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4.When we join AC, we get ∆ABC which is the required triangle.

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Focus on the Steps of Construction if AC > AB:

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

 (Image to be added soon)

2. From the given ray BX you need to cut an arc equal to AB – AC at point P and join it to C. In this case the point P will lie on the opposite side to the ray BX. Now draw the perpendicular bisector of PC and now let it intersect BX at point A as shown below-

(Image to be added soon)

3. Now when we will join the points A and C, the triangle we get as ∆ABC is the required triangle.