Construction of Similar Triangles: Construction of Similar Triangles

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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,

Similar Triangles

Similar Triangles

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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.

Similar Triangles

Similar Triangles

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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).

Similar Triangles

Similar Triangles

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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

Similar Triangles

Similar Triangles

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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.

Construction of Similar Triangles

Construction of Similar Triangles

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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

Construction of Similar Triangles

Construction of Similar Triangles

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We need to construct triangle similar to

Let’s follow these steps

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

Steps of Construction

Steps of Construction

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

Steps of Construction

Steps of Construction

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Thus, is the required triangle.

Justification

Steps of Construction

Steps of Construction

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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

Steps of Construction

Steps of Construction

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(Common)

(From (2))

(AA similarity)

Since corresponding side of similar triangles are in the same ratio

So, (From (1))

Thus, our construction is justified.