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Course: High school geometry>Unit 1

Lesson 2: Introduction to rigid transformations

Identifying transformations

Let's look at four types of transformations: rotations (spinning a shape around a point), translations (shifting a shape), reflections (flipping a shape over a line), and dilations (shrinking or expanding a shape). We practice identifying these transformations in different pairs of figures.

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• What are all the transformations?
• There are four different types of transformations.

Translation:
the object moves up/down/left/right, but the shape of the object stays exactly the same. Every point of the object moves the same direction and distance.

Rotation:
the object is rotated a certain number of degrees about a fixed point (the point of rotation). A positive rotation moves counterclockwise; a negative rotation moves clockwise.

Reflection:
the object is reflected (or "flipped") across a line of reflection, which might be the x-axis, y-axis, or some other line.

Dilation:
the object stays the same shape, but is either stretched to become larger (an "enlargement") or shrunk to become smaller (a "reduction").

It is possible for an object to undergo more than one transformation at the same time.

Hope this helps!
• In the 3rd example, I understand that it is reflection, but couldn't it also be rotation.
• A rotation always preserves clockwise/counterclockwise orientation around a figure, while a reflection always reverses clockwise/counterclockwise orientation.

It can be verified by the distance formula or Pythagorean Theorem that each quadrilateral has four unequal sides (of lengths sqrt(2), 3, sqrt(10), and sqrt(13)). This means there's only one way that the sides of quadrilateral A can correspond to the sides of quadriateral B.

If one travels counterclockwise around the sides of quadrilateral A, then the corresponding sides of quadrilateral B would be in clockwise order. So the transformation reverses clockwise/counterclockwise orientation and therefore cannot be a rotation.

Have a blessed, wonderful day!
• what is dilation
• Dilation makes a triangle bigger or smaller while maintaining the same ratio of side lengths.
• Pause at , it looks like a sting ray!
• Isn't reflection just a rotation?
• Not quite. A reflection is a flip, while a rotation is a turn. Reflections reverse the direction of orientation, while rotations preserve the direction of orientation.

For example, if we list the vertices of a polygon in counterclockwise order, then the corresponding vertices of the image of a reflection are in clockwise order, while the corresponding vertices of the image of a rotation (of the original polygon) are in counterclockwise order.

Have a blessed, wonderful day!
• Can a Dilation be a translation and dilation?
• Yes, a dilation about a point can be expressed as a translation followed by a dilation by the same factor but about a different point.
• So Dilation is when the figure is smaller
• Dilation is when the figure retains its shape but its size changes. This can either be from big to small or from small to big. To dilate a figure, all we have to do is multiply every point's coordinates by a scale factor (>1 for an increase in size, <1 for a decrease). Hope this helps!
• at , sal says the figure has been rotated but I was wondering why it can't be a reflection? both reflection and rotation seem possible, the way I am understanding this.
• If you put an imaginary line in between the two shapes and tried to flip one onto the other, you would not be able to do it without rotating one shape.
• SO does translation and rotation the same
• Translation implies that that every coordinate is moves by (x,y) units
Rotation means that the whole shape is rotated around a 'centre point/pivot' (m).
• whats a reflection
(1 vote)
• Consider a point A and a line k.

If A is on k, then the reflection of A across k is just A itself.

If A is not on k, then the reflection of A across k is the unique point B such that k is perpendicular to segment AB, and k intersects this segment at its midpoint.

Visually, think of the reflection of point A across line k as a mirror image of point A, with line k acting as the mirror.

Have a blessed, wonderful day!