Geometry

Practice flashcards covering geometric transformations including translations, reflections, rotations, dilations, and the properties of congruence and similarity based on the lecture notes.

What is a pre-image?

The original input value/s ($x$) that map to a specific output value ($y$).

What is a translation?

A shape or point that changes its location in given directions (up, down, left, right, or diagonally) where the image does not change size or shape.

What is the formula for determining the image of a translation?

$(x,y) \rightarrow (x + a, y + b)$ (subtract $b$ only if it is negative).

What does a negative value for $a$ or $b$ indicate in a translation?

If $a$ is negative, the movement is to the left; if $b$ is negative, the movement is down.

What are the characteristics of a vertical translation mapping?

The $x$ component is zero, following the formula $(x,y) \rightarrow (x + 0, y + B)$. For example, $(x,y) \rightarrow (x, y - 4)$ results in a translation of $4$ units down.

What are the characteristics of a horizontal translation mapping?

The $y$ component is zero, following the formula $(x,y) \rightarrow (x + a, y + 0)$. For example, $(x,y) \rightarrow (x + 3, y)$ results in a translation of $3$ units right.

What happens to the orientation of vertex points during a reflection?

The orientation is reversed; the pre-image is read clockwise, while the image is read counterclockwise.

What are the coordinate formulas for reflections over the x-axis and y-axis?

For the x-axis: $(x,y) \rightarrow (x, -y)$ (x-coords never change). For the y-axis: $(x,y) \rightarrow (-x, y)$ (y-coords never change).

What is the rule for reflecting a point over the line $y=x$?

Reverse the coordinates. For example, $(3,2) \rightarrow (2,3)$.

Where is an image point located in relation to its pre-image point after a rotation?

Both points are equidistant from the center of rotation.

What is the formula for a $180^{\circ}$ rotation (clockwise or counterclockwise) about the center $(0,0)$?

$$(x, y) \rightarrow (-x, -y)$$

What is the formula for a $90^{\circ}$ clockwise rotation about the center $(0,0)$?

$(x, y) \rightarrow (y, -x)$ (switch coordinates before turning $x$ negative).

What is the formula for a $90^{\circ}$ counterclockwise rotation about the center $(0,0)$?

$(x, y) \rightarrow (-y, x)$ (switch coordinates before turning $y$ negative).

What defines congruent figures?

Figures with the same size and shape, having corresponding angles with the same number of degrees and corresponding sides with the same length.

What is the Transitive Property of Congruence in transformations?

If $ABC = A'B'C'$ and $A'B'C' = A''B''C''$, then $ABC = A''B''C''$ (or $ABC = A'''B'''C'''$).

What is a dilation?

A type of transformation that changes the size of the pre-image to create similar figures.

How is the Scale Factor ($SF$) of a dilation calculated?

$SF = \frac{\text{Image}}{\text{Pre-image}}$ (take the corresponding side from the image for the numerator and the side from the pre-image for the denominator).

How do different scale factor values affect a dilation?

A scale factor less than $1$ is a reduction; greater than $1$ is an enlargement; equal to $1$ makes congruent figures; and equal to $0$ makes no image.

What is the formula for dilation on the coordinate plane?

$(nx, ny)$, where $n = SF$.

How do you find the coordinates of a pre-image using the scale factor?

Divide the coordinates of the image by the scale factor ($SF$).

What are the two requirements for figures to be considered similar?

1. Corresponding angles are congruent. 2. Corresponding sides are proportional.