Before we learn the construction of similar triangle, let us know what similar triangles are

Similar Triangles

If Two triangles âˆ†ABC and âˆ†PQR are said to be similar, following two conditions are satisfied:

1. The corresponding angles of the triangles are equal.

i.e.,

âˆ A = âˆ P, âˆ B = âˆ Q, âˆ C = âˆ R

and

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

That is,

\( \frac {AB}{PQ} \) = \( \frac {BC}{QR} \) = \( \frac {AC}{PR} \)

We write âˆ†ABC and âˆ†PQR are similar as âˆ†ABC~âˆ†PQR

For example;

Consider the two triangles given in the figure,

If âˆ†ABC~âˆ†PQR, What is length of the side PR if AB = 6 cm, AC = 8 cm and PQ = 3 cm?

Since, âˆ†ABC~âˆ†PQR

\( \frac {AB}{PQ} \) = \( \frac {AC}{PR} \)

\( \frac {6}{3} \) = \( \frac {8}{PR} \)

PR = \( \frac {8~Ã—~3}{6} \) = 4 cm

Letâ€™s see how to construct similar triangles.

Consider âˆ†ABC where BC = 6 cm,âˆ B=40Â°and âˆ C=60Â°. Draw a triangle similar to âˆ†ABC 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 âˆ†ABC are twice the sides of âˆ†ABC.

Let âˆ†PQR be the new triangle.

QR = 2 Ã— 6 = 12 cm [Scale factor is 2]

âˆ B = âˆ Q = 40Â°and âˆ C = âˆ R=60Â°.

- Draw QR of length 12 cm
- Draw a line through B which makes an angle of 40Â° from BC.
- Draw a line through C which makes an angle of 60Â° from BC.
- Mark the intersection point of above two lines as P. âˆ†PQR is the required triangle (Refer figure).

Figure 2-How to construct similar triangles

Now, suppose the scale factor is a fraction,like \( \frac 54 \), \( \frac 78 \) etc or suppose we donâ€™t know length of the sides?

Then we wonâ€™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 âˆ†ABC with scale factor \( \frac 35 \).

Scale factor \( \frac 35 \) means, the new triangle will have side lengths \( \frac 35 \) 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 B1, B2, B3, B4 and B5 such that B B1 = B1 B2 = B2 B3 = B4 B5.
- Join B5C
- Draw a line parallel to B5C through B3 [Since 3 is the smallest among 3 and 5] and mark Câ€™ where it intersects with BC.
- Draw a line through the point Câ€™ parallel to AC and mark Aâ€™ where it intersects AB.
- Aâ€™BCâ€™ is the required triangle.

Figure 3-How to construct similar triangles

How can we verify that âˆ†ABC~âˆ†A’BC’ ?

\( \frac {BC’}{C’C}\) = \( \frac 32 \) [By construction]

Therefore,

\( \frac {BC}{BC’}\) = \( \frac {BC’~+~C’C}{BC’}\) = \( 1~+~\frac 23 \) = \( \frac 53 \)

That gives,

\( \frac {BC’}{BC}\) = \( \frac 35 \)<

And, since A’C’ is parallel to AC,

âˆ A’C’B=âˆ ACB [corresponding angles]

âˆ ABC=âˆ A’BC’ [common angle]

Therefore,

âˆ†ABC ~ âˆ†A’BC’

To learn more about triangles, download BYJU’S- The Learning App from Google Play Store.