Basic transformations
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.
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.
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.
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.
Congruence
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.
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.
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.
Similarity
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.
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.