Questions & Answers

Question

Answers

Any two congruent figures are similar.

(A) True

(B) False

Answer
Verified

Hint: In order to solve such problems use the basic definition of similarity and the congruence of the figures. Then find out the relation between the congruent and similar figures. Diagrammatical visualization can also be used.

__Complete step-by-step solution__ -

Let us first understand the basic definition of both similar and congruent figures and their property.

Definition of congruence

Congruent objects on a plane are those that can be transformed one into another by any combination of the following transformations:

- rotation around some point as a center,

- translation (shift) in certain direction,

- symmetry around some axis.

Definition of similarity.

Similar objects on a plane are those that can be transformed one into another by any combination of the following transformations:

- scaling using some point on a plane as a center and some real number as a factor of scaling,

- rotation around some point as a center,

- translation (shift) in certain direction,

- symmetry around some axis.

As we can see, any pair of congruent objects is also similar since all transformations needed for congruence are included into similarity.

Congruent shapes are always similar, but similar shapes are usually not congruent - one is bigger and one is smaller.

Hence, any two congruent figures are always similar.

So option A is correct and the statement is true.

Note: All congruent figures are similar, but not all similar figures are congruent. Congruence means two objects (whether two dimensional or three dimensional) are identical in size and shape. Everything about them - their angles, lengths of sides, overall dimensions - are identical. Similar figures have the same shape and proportions but are not necessarily the same size.

Let us first understand the basic definition of both similar and congruent figures and their property.

Definition of congruence

Congruent objects on a plane are those that can be transformed one into another by any combination of the following transformations:

- rotation around some point as a center,

- translation (shift) in certain direction,

- symmetry around some axis.

Definition of similarity.

Similar objects on a plane are those that can be transformed one into another by any combination of the following transformations:

- scaling using some point on a plane as a center and some real number as a factor of scaling,

- rotation around some point as a center,

- translation (shift) in certain direction,

- symmetry around some axis.

As we can see, any pair of congruent objects is also similar since all transformations needed for congruence are included into similarity.

Congruent shapes are always similar, but similar shapes are usually not congruent - one is bigger and one is smaller.

Hence, any two congruent figures are always similar.

So option A is correct and the statement is true.

Note: All congruent figures are similar, but not all similar figures are congruent. Congruence means two objects (whether two dimensional or three dimensional) are identical in size and shape. Everything about them - their angles, lengths of sides, overall dimensions - are identical. Similar figures have the same shape and proportions but are not necessarily the same size.

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