Finite Mathematics 1: Geometric Dilations Study Guide

Learning Objectives for Geometric Dilations

The study of geometric dilations in Finite Mathematics 1 focuses on several core competencies. Learners are expected to define geometric dilation and describe how the scale factor determines whether a figure is enlarged or reduced. This includes drawing and determining the coordinates of dilated figures on the coordinate plane using a specific center and scale factor. Beyond theoretical application, students must value the importance of proportion and accuracy in resizing figures, relating these mathematical concepts to practical fields such as art, design, layout, and technical drawing.

Exploration and Activities in Geometric Transformation

In Activity 1: Comparing Triangles in a Coordinate Plane, students explore the relationship between sets of coordinates. The activity involves writing the coordinates of vertices for a blue triangle and a red triangle. Students must determine how the coordinate sets are related and how the shapes themselves relate. This exploration extends to drawing a green triangle where every coordinate value is twice that of the corresponding blue triangle, facilitating a discussion on the relationship between original and transformed figures.

Activity 2: Drawing Triangles in a Coordinate Plane requires drawing a triangle with vertices at $(0, 2)$, $(-2, 2)$, and $(1, -2)$. Learners then multiply each coordinate of these vertices by a scale factor of $2$ to obtain three new vertices and observe the resulting relationship between the two triangles. This process is repeated by multiplying coordinate values by a scale factor of $3$.

These activities prompt students to consider how to enlarge or reduce a figure in the coordinate plane in their own words. Furthermore, they are asked to describe the importance of knowing these procedures in professional careers like drafting, where technical drawings frequently require precise resizing.

Formal Definition of Dilation

A dilation is a type of transformation in which the preimage and its final image are similar. It is classified as a similarity transformation because it preserves angle measure, the betweenness of points, and collinearity. However, unlike rigid transformations, a dilation does not preserve distance.

A dilation with a specific center $C$ and a scale factor $k$ maps every point $P$ in a figure to a corresponding image point $P'$ based on the following criteria:

1. If point $P$ is not the center point $C$, then the image point $P'$ lies on the ray $CP$. In this scenario, the scale factor $k$ is a positive number such that $k = \frac{CP'}{CP}$ and $k \neq 1$.

2. If point $P$ is the center point $C$ themselves, then the point remains unchanged during the transformation, such that $P = P' bills$.

Classification of Dilations: Reduction vs. Enlargement

Dilations are categorized based on the value of the scale factor $k$:

• An enlargement occurs when a dilation preimage creates a larger image. This happens when the scale factor is greater than one ($k > 1$).

• A reduction occurs when a dilation preimage creates a smaller or reduced image. This happens when the scale factor is between zero and one ($0 < k < 1$).

Because the resulting figures are similar ($\Delta PQR \sim \Delta P'Q'R'$), the ratio of matching sides, such as $\frac{P'Q'}{PQ}$, is equal to the scale factor of the dilation.

Coordinate Plane Rules and Examples

In a coordinate plane, when the center of the dilation is the origin $(0, 0)$, the image of a point $P(x, y)$ is found by multiplying both the $x$ and $y$ coordinates by the scale factor $k$, resulting in $P'(kx, ky)$.

Example 1: Identification and Calculation

• Part A: Given distances from the center $C$, where $CP' = 2$ and $CP = 3$, the scale factor is $k = \frac{CP'}{CP} = \frac{2}{3}$. Since $k < 1$, this is a reduction.

• Part B: Given distances where $CP' = 2$ and $CP = 1$, the scale factor is $k = \frac{CP'}{CP} = \frac{2}{1} = 2$. Since $k > 1$, this is an enlargement.

Example 2: Dilating a Rectangle

Consider a rectangle $ABCD$ with vertices at $A(2, 2)$, $B(6, 2)$, $C(6, 4)$, and $D(2, 4)$. If we use the origin as the center and a scale factor of $k = \frac{1}{2}$, the new coordinates are calculated as follows:

• $A(2, 2) \rightarrow A'(1, 1)$
• $B(6, 2) \rightarrow B'(3, 1)$
• $C(6, 4) \rightarrow C'(3, 2)$
• $D(2, 4) \rightarrow D'(1, 2)$

In this scenario, because the scale factor is $\frac{1}{2}$, every linear dimension of the image is half that of the preimage, meaning the perimeter of the image is exactly half of the perimeter of the preimage.

Methods of Solving Dilations

There are three primary methods used to solve or describe dilations:

Method 1: Graphical Solution Methods

• Manual Measurement: This involves drawing a line from the dilation center point to each preimage vertex. Using a ruler, measure the distance from the center point to each vertex. Multiply these distances by the Scale Factor ($SF$) to find the length and location of each image vertex. Example: A reduction of $\frac{1}{2}$ where a preimage line $OB$ measuring $2.5\,inches$ results in an image line $OB'$ measuring $1.25\,inches$.
• Grid Blocks: For transformations on a grid, count the horizontal and vertical grid blocks ($\Delta x$ and $\Delta y$) from the center to each preimage vertex. Multiply these counts by the scale factor to determine the grid position of the image vertices.

Method 2: Symbol or Notation Method

• The notation for a dilation about the origin with a scale factor $SF$ is $D_{O, SF}$.
• If the center is an $(x, y)$ coordinate other than the origin, the notation is $D_{(x,y), SF}$.
• The general functional notation is $D_{O, k} \Delta ABC(x, y) \rightarrow \Delta A'B'C'(k \cdot x, k \cdot y)$.

Method 3: Verbal Description Method

• This simply describes the transformation in words. For example: "Dilate (enlarge) the preimage by an $SF = 3$ with the center point of the dilation at the origin." In this case, each image vertex coordinate is exactly three times its corresponding preimage coordinate.

Real-World Application: Shadow Puppets

Shadow puppets illustrate the principle of geometric dilation. A flat figure is held between a light source and a screen. The light source acts as the center of dilation, and the shadow seen by the audience is an enlarged image of the puppet. This relationship follows the rule of triangle similarity, where $\Delta LCP \sim \Delta LSH$ ($L$ is the light, $C$ and $P$ are points on the puppet, and $S$ and $H$ are points on the shadow).

Problem Setup: A shadow puppet is $12\,inches$ tall ($CP = 12\,in$). We determine the height of the shadow ($SH$) based on different distances.

Case A: $LC = LP = 59\,in$ and $LS = LH = 74\,in$
Using the ratio $\frac{LC}{LS} = \frac{CP}{SH}$:
$\frac{59}{74} = \frac{12}{SH}$
$59 \cdot SH = 12 \cdot 74$
$59 \cdot SH = 888$
$SH \approx 15\,inches$
Scale factor calculation: $\frac{SH}{CP} = \frac{15}{12} = 1.25$. The shadow is $25\%$ larger than the puppet.

Case B: $LC = LP = 66\,in$ and $LS = LH = 74\,in$
Using the ratio $\frac{LC}{LS} = \frac{CP}{SH}$:
$\frac{66}{74} = \frac{12}{SH}$
$66 \cdot SH = 888$
$SH \approx 13.45\,inches$
Scale factor calculation: $\frac{SH}{CP} = \frac{13.45}{12} = 1.12$. The shadow is $12\%$ larger than the puppet.

Vocabulary

• Image: The final shape and location of the figure after it has been transformed.
• Preimage: The initial or original shape and location of the figure prior to transformation.
• Dilation: The enlargement or reduction in the size of a shape from its preimage position to its image position.
• Scale Factor: The numerical multiplier amount that either enlarges or reduces the preimage size to create the image size.