# Construction of Similar Triangles: Construction of Similar Triangles (For CBSE, ICSE, IAS, NET, NRA 2022)

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What similar triangles are

## Similar Triangles

If Two triangles and are said to be similar, following two conditions are satisfied:

1. The corresponding angles of the triangles are equal.

i.e.. ,

and

2. Since, and are two similar triangles, their corresponding sides are in a ratio or proportion.

That is,

We write and are similar as

For example;

Consider the two triangles given in the figure,

If , What is length of the side PR if and

Since,

let՚s see how to construct similar triangles.

Consider where and Draw a triangle similar to with a scale factor 2.

Scale factor is the ratio of the sides of the triangle to be constructed with the corresponding sides of the given triangle.

Figure 1-Triangle ABC

Here, a scale factor of 2 means that sides of the new triangle which is similar to ∆ ABC are twice the sides of ∆ ABC.

Let be the new triangle.

[Scale factor is 2]

• Draw QR of length
• Draw a line through B which makes an angle of from BC.
• Draw a line through C which makes an angle of from BC.
• Mark the intersection point of above two lines as P. ∆ PQR is the required triangle (Refer figure) .

Figure 2-How to construct similar triangles

• Now, suppose the scale factor is a fraction, like etc or suppose we don՚t know length of the sides?
• Then we won՚t be able to construct similar triangles precisely.
• The method to construct a similar triangle precisely is discussed here.
• Problem: Construct a triangle which is similar to ∆ ABC with scale factor

Scale factor means, the new triangle will have side lengths times the corresponding side lengths.

## Construction of Similar Triangles

• The steps of construction of similar triangles are as follows (Refer figure)
• Draw a ray BX which makes acute angle with BC on the opposite side of vertex A.
• Locate 5 points on the ray BX and mark them as and such that B
• Join
• Draw a line parallel to through [Since 3 is the smallest among 3 and 5] and mark C ′ where it intersects with BC.
• Draw a line through the point C ‘parallel to AC and mark A’ where it intersects AB.
• is the required triangle.

Figure 3-How to construct similar triangles

How can we verify that ?

Therefore,

That gives,

And, since is parallel to AC,

Therefore,

Question:

Construct a triangle similar to a given with its sides equal to of the corresponding sides of the (Scale factor ) .

Here, we are given , and scale factor

Scale factor

We need to construct triangle similar to

### Steps of Construction

1. Draw any ray making an acute angle with BC on the side opposite to the vertex A.

2. Mark 5 (the greater of 5 and 3 in ) points.

on BX so that

3. Join (3rd point as 3 is smaller in ) and draw a line through parallel to , to intersect BC extended at .

Thus, is the required triangle.

Justification

Since scale factor is ,

We need to prove

By Construction,

Also, is parallel to AC

SO, they will make the same angle with line BC

Now,

In and

(Common)

(From (2) )

(AA similarity)

Since corresponding side of similar triangles are in the same ratio

So, (From (1) )

Thus, our construction is justified.

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