Geometry Chapter 11

Isometry

Isometry

A transformation that does not change the shape or size of a figure

Types of Isometries

Reflection, translation, and rotation

Other Names for Isometries

Congruence transformation or rigid motions

Preimage

The original figure

Image

Resulting figure

Reflection

A transformation across a line which is the perpendicular bisector of each segment joining each point and its image

Reflection across the x-axis

A(x,y)→A’(x,-y)

Reflection across the y-axis

A(x,y)→A’(-x,y)

Reflection across the line y=x

A(x,y)→A’(y,x)

Translation

A transformation where all the points of a figure are moved the same direction. The image is congruent to the preimage and when transformed along a vector the image has the same length as the vector and is parallel to the vector.

Translation along a Vector

A(x,y)→A’(x+a,y+b)

Rotation

A transformation that turns a figure around a fixed point.

Center of Rotation

The fixed point around which a rotation is made

Line of Reflection

A straight line across which a reflection is made

Rotation by 90 about the Origin

A(x,y)→A’(-y,x)

Rotation by 180 about the Origin

A(x,y)→A’(-x,-y)

Composition of Transformation

One transformation followed by another

Glide Reflection

The composition of a translation and a reflection across a line parallel to the translation vector

Isometry

A composition of two isometries is a(n) _______.

Translation

The composition of two reflections across two parallel lines is equivalent to a(n) _______.

Rotation

The composition of two reflections across two intersecting lines is equivalent to a(n) __________.

Reflections

Any translation or rotation is equivalent to a composition of two ___________.

Symmetry

A figure has __________ if there is a transformation of the figure such that the image coincides with the preimage

Line Symmetry

A figure has ______ if it can be reflected across a line so that the image coincides with the preimage

Line of Symmetry

Divides the figure into two congruent halves

Rotational Symmetry

A figure has ___________ if it can be rotated about a point by an angle greater than 0 and less than 360 so that the image coincides with the preimage

Plane Symmetry

A three-dimensional figure has ________ if a plane can divide the figure into two congruent reflected halves