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## #1 2024-05-19 17:45:46

Jai Ganesh
Registered: 2005-06-28
Posts: 47,795

### Similarity (geometry)

Similarity (geometry)

In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other.

For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other. This is because two ellipses can have different width to height ratios, two rectangles can have different length to breadth ratios, and two isosceles triangles can have different base angles.

If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure.

Two congruent shapes are similar, with a scale factor of 1. However, some school textbooks specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #2 2024-05-19 22:53:32

KerimF
Member
From: Aleppo-Syria
Registered: 2018-08-10
Posts: 225

### Re: Similarity (geometry)

If I understood well; to these schools, two congruent triangles, with a scale factor of 1, are not similar. They likely added the notion of 'identical' to be used instead of 'similar', in this case.

Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).
But only a human may have the freedom and ability to oppose his natural robotic nature.
But, by opposing it, such a human becomes no more of this world.

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## #3 2024-05-20 20:40:59

Jai Ganesh
Registered: 2005-06-28
Posts: 47,795

### Re: Similarity (geometry)

I used the source Wikipedia.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #4 2024-08-22 21:18:22

Ibrahim Ali
Member
Registered: 2020-05-27
Posts: 3

### Re: Similarity (geometry)

Similarity means shapes that look the same even if they're resized, rotated, or flipped. So, all squares or circles are similar, but different ellipses or rectangles aren’t.

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