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  1. Home
  2. 12th grade
  3. Geometry
  4. Course : Transformations, congruence and similarity

Transformations, congruence and similarity Geometry

Summary

IBasic transformationsAReflectionBRotationCTranslationDDilationIICongruenceACongruent segmentsBCongruent anglesCCongruent figuresIIISimilarityADefinitionBSimilar triangles
I

Basic transformations

A

Reflection

Reflection

A reflection of a geometric object is the mirror image of an object across a line.

In the following graphic, a triangle is reflected across a line.

-

Reflecting a figure preserves all of its lengths and angles.

B

Rotation

Rotation

A rotation of a geometric object is the image of a geometric object after being turned around a fixed point.

In the following graphic, a geometric object is rotated around a fixed point.

-

Rotating a figure preserves all of its lengths and angles.

C

Translation

Translation

A translation of a geometric object is the image of a geometric object after being moved around without any rotations or reflections.

In the following graphic, a geometric object has been translated.

-

Translating a figure preserves all of its lengths and angles.

D

Dilation

Dilation

A dilation of a geometric object is obtained by shrinking or enlarging the geometric object. The shape of a dilated object is the same as the original, but the size of the object is different.

In the following graphic, a geometric object in blue has been dilated to the geometric object in red.

-
II

Congruence

A

Congruent segments

Congruent segments

Two line segments of equal length are said to be congruent segments. Congruent segments are indicated by drawing the same amount of little tic lines on the segments.

In the following graphic, several pairs of congruent segments are given.

-

Observe that two line segments can be congruent without being parallel.

B

Congruent angles

Congruent angles

Two angles are said to be congruent if they have the same measure.

In the following graphic, two pairs of congruent angles are given.

-

Observe that two angles can be congruent without having sides that are parallel or congruent.

C

Congruent figures

Congruent figures

Two geometric objects are said to be congruent if they have the same size and shape.

The following graphic contains two geometric objects which are congruent.

-

If two objects are congruent, then they have sides which are congruent to one another and corresponding angles which are also congruent to one another.

Moreover, if two objects are congruent, then one figure can be obtained from the other via a sequence of rotations, reflections, and translations.

III

Similarity

A

Definition

Similarity

Two geometric objects are said to be similar if one object can be obtained from the other via a sequence of transformations involving reflections, rotations, translations, and dilations.

The following graphic contains two geometric objects which are similar.

-
B

Similar triangles

Corresponding angles of similar triangles

If two triangles are similar, then the corresponding angles are congruent.

The following graphic contains two similar triangles.

-

Because the two triangles are similar:

  • \alpha=\alpha_1
  • \beta=\beta_1
  • \gamma=\gamma_1

Corresponding sides of similar triangles

If two triangles are similar, then the ratios of corresponding sides agree.

The following graphic contains two similar triangles.

-

Because the two triangles are similar:

\dfrac{A}{a}=\dfrac{B}{b}=\dfrac{C}{c}

The following graphic contains two similar triangles.

-

The previous theorem allows us to compute the lengths x and y. Specifically, we know the following:

\dfrac{6}{2}=\dfrac{y}{3}=\dfrac{x}{1}

Therefore:

  • y=9
  • x=3

We have found the missing lengths of the similar triangle.

See also
  • Exercise : Identify transformations
  • Exercise : Graph transformations
  • Exercise : Determine whether line segments are congruent
  • Exercise : Determine whether figures are congruent
  • Exercise : Use the properties of similar triangles to determine measures (cos, sin, tan, lengths, angles)
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