Unit 01: Shapes & Transformations

Good Flow

● Clear pathways and doorways ● Maximizes open space ● Ability to access each part of the room

Flow

The ability to move steadily and continuously in a pathway.

Rigid Motion

The combination of transformations when the pre-image is congruent to the image.

Vertex (Vertices)

(a) For a two-dimensional geometric shape, a point where two or more line segments or rays meet to form a "corner," such as in a polygon or angle. (b) For a three-dimensional polyhedron, a point where the edges of the solid meet. (c) On a graph, it can be used to describe the highest or lowest point on the graph of a parabola or absolute value function (depending on the graph's orientation).

Ratio

A comparison of two quantities by division. Can be written using a colon, but is more often written as a fraction. For example, in the two similar triangles below, a __________ can be used to compare the length of BC in ΔABC with the length of EF in ΔDEF. This __________ can be written as 5:11 or as the fraction .

Prove (proof)

A convincing logical argument that uses definitions and previously proven conjectures in an organized sequence. (To show that that a conjecture is true. It be written in a paragraph, represented with a flowchart, or documented in a formal two-column proof)

Line of Symmetry

A line that divides a shape into two congruent parts that are mirror images of each other.

Reflection

A mirror image where the figure does not have the same orientation as the pre-image because the order of the vertices will be different, figures can be a reflected over axes and a given linear equation

Midpoint

A point that divides a segment into two segments of equal length.

Regular Polygon

A polygon is regular if it is a convex polygon with congruent angles and congruent sides. For example, the shape at right is a regular hexagon.

Slope

A ratio that describes how steep (or flat) a line is. It can be positive, negative, or even zero, but a straight line has only one _________.

Translation

A shift or a slide of a figure vertically, horizontally, or diagonally, where the image will always have the same orientation as the pre-image.

Isometry

A transformation in which the pre-image and image are congruent.

Angle Preserving

A transformation is said to be angle preserving if (1) the image of any angle is again an angle and (2) for any given angle, the angle measure of the image of that angles is equal to the angle measure of the pre-image of that angle.

Distance Preserving

A transformation is said to be distance preserving if the distance between the images of two points is always equal to the distance between the pre-images of the two points.

Isosceles Triangle

A triangle with two congruent sides, and two congruent angles (these angles are called based angles, the remaining angle is call the vertex angle).

Scalene Triangle

A triangle with zero sides congruent (all sides have different lengths)

Investigative process

A way to study and learn new mathematical ideas. Mathematicians have used this process for many years to make sense of new concepts and to broaden their understanding of older ideas.

Straight Angle

An angle that measures 180°. This occurs when the rays of the angle point in opposite directions, forming a line.

Right Angle

An angle that measures 90°. A small square is used to note a _________________.

Circular Angle

An angle with a measure of 360°.

Acute Angle

An angle with measure greater than 0° and less than 90°.

Conjecture

An educated guess; often resulting from noticing a pattern during an investigation. ___________ are also often written in conditional ("If..., then...") form. Once a ______________ is proven, it becomes a theorem.

Obtuse Angle

Any angle that measures between (but not including) 90° and 180°.

Equidistant

At equal distances.

Polygon

Closed figure with straight sides

Congruent Figures

Have corresponding sides that are congruent and corresponding angles that are congruent

How do you determine if a figure has reflectional symmetry or a line of symmetry?

If you fold a shape over a line between vertices, the shapes on both sides of the line will match each other perfectly. Some shapes have more than one or no lines of symmetry.

Does the pre-image and image have the same orientation when reflected?

No

Area

On a flat surface (plane), the number of non-overlapping square units needed to cover the region

Mapped

Refer to functions that take points on a line or in the plane, to other points on a line or in the plane.

Orientation

Refers to the placement and alignment of a figure in relation to others or an object of reference (such as coordinate axes). Unless it has rotation symmetry, the _______________ of a figure changes when it is rotated (turned) less than 360°. Also, the __________________ of a shape changes when it is reflected, except when reflected across a line of symmetry.

measure of an angle

Represents the number of degrees of rotation from one ray to the other about the vertex.

line of reflection

Same as the line of symmetry.

Perimeter

The distance around the exterior of a figure on a flat surface. For a polygon, the ____________ is the sum of the lengths of its sides. The _______________ of a circle is also called a circumference.

Image

The figure as a result of a transformation

Pre-image

The figure before it is transformed or also called the original.

Rotation

The position of the figure is turned and will have the same orientation from the pre-image, can be turned 90º, 180º, 270º, 360º, clockwise or counterclockwise.

What are the types of transformations?

Translations, reflections, rotations are transformations.

Equilateral Triangle

Triangle with all sides congruent (and all angles aka equiangular)

Because of rigid motions, we know...

Two figures are congruent if and only if there is a sequence of reflections, translations, and/or rotations that maps one figure onto the other.

Parallel Lines

Two lines on a flat surface that never intersect. There is a constant distance (equidistant) between the two lines (or line segments).

Perpendicular

Two rays, line segments, or lines that meet (intersect) to form a right angle (90º).

Transformation

When a figure is changed in position and/or size in the coordinate plane.

Reflection Symmetry

When a shape appears not to change after being reflected across a line.

Rotational Symmetry

When a shape can be rotated for less than 360° and it appears not to change.

Corresponding

When sides and angles match or are in the same location in two or more figures

Does the pre-image and image have the same orientation when rotated?

Yes

Does the pre-image and image have the same orientation when translated?

Yes

Symmetry

having the same shape, size, and position on both sides of a dividing line; balanced proportions