Geometric Translations and Vector Components Study Notes

Spiritual Foundation and Opening Prayer

The lesson on geometric translations begins with a spiritual invocation, seeking blessings and guidance for students, families, and teachers. The prayer uses mathematical metaphors to express faith: the sum of gifts is asked to multiply God's goodness, and faith is requested to subtract sins. The goal is to remain in a state of grace through the guidance of the Lord, acting as a "multiplier of goodness" while preventing evil from becoming a "divisor of all men."

Daily Numeracy Review and Math Refresh

The math refresh session covers foundational algebraic and numeric concepts essential for advanced geometry. Five specific problems define the Day 1 review for Week 2.

Problem 1: Factors and Sums. A riddle asks for two numbers that add to the bottom number but multiply to the top number in a cross-diagram. Given the top number is $15$ and the bottom number is $8$, the solution identifies the numbers as $3$ and $5$, since $3 \times 5 = 15$ and $3 + 5 = 8$.

Problem 2: Word Problems. Students are asked to find a number based on the following description: "Two less than $2$ times a number is the same as the number plus $64$." This is represented by the equation $2x - 2 = x + 64$. Solving for $x$ involves subtracting $x$ and adding $2$ to both sides, resulting in $x = 66$.

Problem 3: Verbal Expressions. The task is to write a mathematical expression matching the phrase "seven less than a number." The resulting expression is $x - 7$.

Problem 4: Inequalities. The phrase "The sum of a number and $7$ is no more than $53$ " must be translated into an inequality. The correct representation is $x + 7 \le 53$.

Problem 5: Solving for Variables. Given the equation $m - n = 5$, students are tasked with solving for $m$. By isolating the variable, the solution is $m = n + 5$.

Review of Rotational Transformations

Before diving into translations, a review of rotations on the coordinate plane is conducted to ensure prerequisite skills are in place.

Question 1: Coordinate Rotation. Find the coordinates of point $S$ after a $90^{\circ}$ counterclockwise rotation about the origin. If original point $S$ is at $(3, 3)$, its image $S'$ after a $90^{\circ}$ counterclockwise rotation becomes $(-3, 3)$.

Question 2: Triangle Vertices. Triangle $\Delta NPQ$ has vertices $N(-6, -4)$, $P(-3, 4)$, and $Q(1, 1)$. Rotating the triangle $90^{\circ}$ counterclockwise about the origin maps the vertex $P(-3, 4)$ to the point $P'(-4, -3)$.

Question 3: Identifying Rotation Degrees. Students must look at a provided figure to determine the degree of rotation. If a figure moves from Quadrant I to Quadrant IV in a specific orientation, it represents a $90^{\circ}$ clockwise rotation.

Question 4: Mapping 270 Degrees. Triangle A is rotated $270^{\circ}$ counterclockwise about the origin. The review asks which triangle (B, C, or D) represents the new location. The correct location for a $270^{\circ}$ counterclockwise rotation matches the position of Triangle B.

Question 5: Transformation of Multiple Vertices. Vertices $L(0, 3)$, $K(-3, 2)$, and $M(-2, 4)$ undergo a $90^{\circ}$ counterclockwise rotation. The resulting coordinates are $L'(-3, 0)$, $K'(-2, -3)$, and $M'(-4, -2)$.

Introduction to Geometric Translations

Geometric transformations include translation, reflection, rotation, dilation, and glide reflection. This session focuses specifically on the meaning and properties of geometric translation. The learning objectives include describing translations, performing translations on a coordinate plane using multiple notations (words, coordinate notation, and vector form), and recognizing these movements in real-life designs and mechanics.

Real-World Scenario: The Airport Transfer Challenge

To conceptualize translations, the "Airport Transfer Challenge" is introduced. Abigail uses three mechanical devices at an airport, each modeling a translation in a straight line:

1. Elevator: Abigail enters at the fourth floor and descends to the first floor. This represents a vertical translation downward.
2. Escalator: Abigail moves from the first floor to the second floor. This represents a translation that is both horizontal and vertical.
3. Moving Walkway: Abigail moves from Gate 1 to Gate 17. This represents a horizontal translation.

In each situation, Abigail moves from point $A$ (preimage) to point $A'$ (image). Her position changes, but her physical attributes (size, shape, orientation) remain constant. In geometry, this sliding movement is the definition of a translation.

Defining Properties of Translations

A translation is a movement that slides a figure to a new position without changing its size, shape, or orientation. This means the figure is not flipped or rotated. The resulting image is always congruent to the original figure (preimage). Specifically, if a translation maps points $P$ and $Q$ to points $P'$ and $Q'$, then the distances $PP'$ and $QQ'$ are equal ($PP' = QQ'$) and the segments $PP'$ and $QQ'$ are either parallel ($PP' \parallel QQ'$) or collinear.

Methods of Representing Translations

There are three primary methods to describe a translation:

Method 1: Verbal Description Method. This uses words to specify direction and distance. For example: "Translate the figure $3 \, \text{units}$ to the right and $6 \, \text{units}$ downward."

Method 2: Coordinate Mapping Method. This uses the notation $(x, y) \rightarrow (x + a, y + b)$, where $a$ and $b$ are constants. $a$ represents the horizontal shift and $b$ represents the vertical shift.

Method 3: Symbol or Notation Method (Vector Form). This uses the symbol $T_{x,y}$ or component form brackets $\langle a, b \rangle$. A vector is a quantity with both direction and magnitude, represented by an arrow from an initial point (start) to a terminal point (end).

Example Problems for Coordinate Mapping

Example 1: Translate a triangle $3 \, \text{units}$ right and $3 \, \text{units}$ down. If the original vertices are given, the new coordinates are found by adding $3$ to each $x$-coordinate and subtracting $3$ from each $y$-coordinate. The resulting image points are $A'(1, -2)$, $B'(5, 2)$, and $C'(4, -1)$.

Example 2: Quadrilateral $ABCD$ has vertices $A(-1, 2)$, $B(-1, 5)$, $C(4, 6)$, and $D(4, 2)$. The translation is defined as $(x, y) \rightarrow (x + 3, y - 1)$. The new vertices are:

• $A(-1, 2) \rightarrow A'(2, 1)$
• $B(-1, 5) \rightarrow B'(2, 4)$
• $C(4, 6) \rightarrow C'(7, 5)$
• $D(4, 2) \rightarrow D'(7, 1)$

Translation Using Vectors

A vector is named by its initial point and terminal point, such as vector $PQ$. The component form of a vector combines the horizontal and vertical components. If point $P$ moves $5 \, \text{units}$ horizontally and $3 \, \text{units}$ vertically to reach point $Q$, the component form is $\langle 5, 3 \rangle$.

Example 3: Identifying vector components. If vector $JK$ moves $3 \, \text{units}$ right and $4 \, \text{units}$ up, its form is $\langle 3, 4 \rangle$. If vector $MN$ moves only vertically by $4 \, \text{units}$, its form is $\langle 0, 4 \rangle$. If vector $TS$ moves $3 \, \text{units}$ right and $3 \, \text{units}$ down, its form is $\langle 3, -3 \rangle$.

Example 4: Mapping a quadrilateral. A quadrilateral $QRST$ maps to $Q'R'S'T'$. By choosing vertex $R$ and its image $R'$, we observe a movement of $8 \, \text{units}$ left and $2 \, \text{units}$ up. The component form is $\langle -8, 2 \rangle$.

Formal Theorems of Translation

Theorem 7.4 (Translation Theorem): A translation is an isometry. An isometry is a transformation that preserves distance, meaning the preimage and image are congruent.

Theorem 7.5 (Double Reflection Theorem): If lines $k$ and $m$ are parallel, then a reflection in line $k$ followed by a reflection in line $m$ results in a translation. If $P''$ is the double-reflected image of $P$, the following properties hold:

1. The segment $PP''$ is perpendicular to both lines $k$ and $m$.
2. The distance of the translation is twice the distance between the parallel lines. If $d$ is the distance between $k$ and $m$, then $PP'' = 2d$.

Application of Theorem 7.5

In a diagram where line $k \parallel m$, reflection in $k$ maps $GH$ to $G'H'$, and reflection in $m$ maps $G'H'$ to $G''H''$. Given $HB = 5$ (distance from point to line $k$) and $DH'' = 2$ (distance from image to line $m$):

• Congruent segments: $GH \ncong G'H' \cong G''H''$; also $HB \cong H'B$ and $H'D \cong H''D$.
• Rectangular Logic: The distance $AC$ equals $BD$ because they are opposite sides of a rectangle formed between the parallel lines.
• Length of translation: The total length of the shift $GG''$ is equal to $HH''$. This is calculated as $2(HB) + 2(DH'') = (5 + 5) + (2 + 2) = 14 \, \text{units}$.

Advanced Independent Practice and Algebra

Students are tasked with solving algebraic problems involving translations. For instance, if a translation by vector $\langle 4, 1 \rangle$ maps point $A(-1, w)$ to $A'(2x+1, 4)$, the following equations are solved:

• For $x$: $-1 + 4 = 2x + 1 \Rightarrow 3 = 2x + 1 \Rightarrow 2 = 2x \Rightarrow x = 1$.
• For $w$: $w + 1 = 4 \Rightarrow w = 3$.

If the same vector maps $B(8y-1, 1)$ to $B'(3, 3z)$, the solutions are:

• For $y$: $8y - 1 + 4 = 3 \Rightarrow 8y + 3 = 3 \Rightarrow 8y = 0 \Rightarrow y = 0$.
• For $z$: $1 + 1 = 3z \Rightarrow 2 = 3z \Rightarrow z = \frac{2}{3}$.

Real-World Examples: Navigation and Design

Navigation Example: A boat travels from island $A(0,0)$ toward island $D$. It reaches point $B(3,2)$ when a storm pushes it to point $C(4,2)$. The destination $D$ is at $(8, 4.5)$.

• Vector $AB$: $\langle 3, 2 \rangle$
• Vector $BC$: $\langle 1, 0 \rangle$
• Required Vector $CD$ to reach destination: $\langle 8-4, 4.5-2 \rangle = \langle 4, 2.5 \rangle$.

Design/CAD Example: Mosaic floors often use translations to repeat patterns. If a square design $A$ at $(0,0)$ is shifted to square $B$ via the mapping $(x, y) \rightarrow (x + 6, y)$, the design uses a horizontal translation of $6$. To map square $A$ to square $C$, which is located directly above $A$ at $(0, 6)$, the mapping is $(x, y) \rightarrow (x, y + 6)$.

Questions & Discussion

Question: A hiker travels $3 \, \text{miles}$ East and $2 \, \text{miles}$ North. What is the component form? Response: The movement East corresponds to the $x$-axis and North to the $y$-axis, resulting in the vector $\langle 3, 2 \rangle$.

Question: Error Analysis. Jerome states that the vector $PQ$ connecting $P(0,0)$ to $Q(-6, 2)$ is $\langle -6, 2 \rangle$. Is Jerome correct? Response: Yes, the vector component is found by subtracting the initial coordinates from the terminal coordinates: $\langle -6 - 0, 2 - 0 \rangle = \langle -6, 2 \rangle$.

Question: How does translation apply to "opening a window"? Response: In a double-hung window (sliding up/down) or a sliding window (sliding left/right), the movement is a translation because the window pane maintains its shape and orientation. A casement window (using a hinge) is a rotation, not a translation.