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Triangles

    Similar Figures

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10th class > Triangles > Similar Figures

Similar Figures

Now consider any two circles from the figure.

Are they congruent?

Note that some are congruent and some are not, but all of them have the same shape. So they all are, what we call, similar. Two similar figures have the same shape but not necessarily the same size. Therefore, all circles are .

Here also all squares are similar and all equilateral triangles are similar.

From the above, we can say that all congruent figures are similar but the similar figures need not be congruent.

Fig A
Fig B
Fig C

You will at once say that they are the photographs of the same monument (Taj Mahal) but are in different sizes. Would you say that the three photographs are similar? .

What can you say about the two photographs of the same size of the same person one at the age of 10 years and the other at the age of 40 years? Are these photographs similar? These photographs are of the same size but certainly they are not of the same shape. So, they are .

This really means that every line segment of the smaller photograph is enlarged (increased) in the ratio 35:45 (or 35:55). It can also be said that every line segment of the bigger photograph is reduced (decreased) in the ratio 45:35 (or 55:35). Further, if you consider inclinations (or angles) between any pair of corresponding line segments in the two photographs of different sizes, you shall see that these inclinations(or angles) are always .

Note that the same ratio of the corresponding sides is referred to as the scale factor (or the Representative Fraction) for the polygons. You must have heard that world maps (i.e., global maps) and blue prints for the construction of a building are prepared using a suitable scale factor and observing certain conventions.

Place a lighted bulb at a point O on the ceiling and directly below it a table in your classroom. Let us cut a polygon, say a quadrilateral ABCD, from a plane cardboard and place this cardboard parallel to the ground between the lighted bulb and the table. Then a shadow of ABCD is cast on the table. Mark the outline of this shadow as A' B' C' D'(in figure).

Note that the quadrilateral A' B' C' D' is an enlargement (or magnification) of the quadrilateral ABCD. This is because of the property of light that light propogates in a straight line. You may also note that A' lies on ray OA, B' lies on ray OB, C lies on OC and D' lies on OD.

Thus, quadrilaterals A' B' C' D' and ABCD are of the same shape but of different sizes.

(i) ∠A =∠A' ∠B=∠B' ∠C=∠C' ∠D=∠D' and

(ii) ABA'B' = BCB'C' = CDC'D' DAD'A'

From the below figure, you can easily say that quadrilaterals ABCD and PQRS of are .

From the Fig, you can easily say that quadrilaterals ABCD and PQRS of are .

Remark : You can verify that if one polygon is similar to another polygon and this second polygon is similar to a third polygon, then the first polygon is similar to the third polygon.

You may note that in the two quadrilaterals (a square and a rectangle) of in the figure: the corresponding angles are , but their corresponding sides are not in the ratio.

So, the two quadrilaterals are not . Similarly, you may note that in the two quadrilaterals (a square and a rhombus) of the above figure, the corresponding sides are in the same ratio, but their corresponding angles are not equal. Again, the two polygons (quadrilaterals) are not .