Scaling the World Slides Geometry: Transformations
Scaling the World
Introduction to Dilations and Scale Factors
The Shadow Effect
Imagine you are making shadow puppets with a flashlight.
"What happens to the size of the shadow as you move your hand closer to the flashlight?"
The hand is the Pre-Image.
The shadow is the Image.
The flashlight is the Center of Dilation.
What is a Dilation?
A Dilation is a transformation that changes the size of a figure but not its shape.
Center of Dilation
The fixed point about which a figure is enlarged or reduced.
Scale Factor (\(k\))
The ratio of the lengths of corresponding sides.
Types of Dilations
Enlargement
\(k > 1\)
The image is larger than the pre-image. It moves further away from the center.
Reduction
\(0 < k < 1\)
The image is smaller than the pre-image. It moves closer to the center.
Calculating \(k\)
The Golden Rule
\[k = \frac{\text{Image Length}}{\text{Pre-Image Length}}\]
Always "New" over "Old"
Distance from center scales too!
Dilation Discovery Worksheet Dilation Discovery
Geometry • Similarity Unit
Name
Date
Goal: Measure pre-images and images to determine the relationship between scale factors and side lengths.
1
Shadow Puppet Scenario
A student creates a shadow puppet using a flashlight. The hand (Pre-Image) is placed 10 inches from the flashlight (Center). The shadow (Image) appears on a wall 40 inches from the flashlight.
A. Calculate the Distance Scale Factor:
Distance from Center to Image / Distance from Center to Pre-Image
B. Predicted Size:
If the hand is 4 inches tall, how tall will the shadow be?
Diagram: Light Source (C) -> Hand (P) -> Shadow (I)
2
Measurement Matrix
Use your ruler to measure the lengths provided in centimeters. Calculate the scale factor (k).
Figure Case Pre-Image Length Image Length Scale Factor (k) Case Alpha: Square 2 cm 6 cm Case Beta: Triangle 12 cm 3 cm Case Gamma: Circle (Rad.) 5 cm 7.5 cm
3
Properties of Dilations
1. Does a dilation change the measure of the angles in a figure? Explain based on your observations.
2. If a scale factor is \(k = 1\), what happens to the image? Is it an enlargement or a reduction?
3. A triangle with a perimeter of 15 cm is dilated with \(k = 0.4\). What is the perimeter of the image?
Module: Non-Rigid Motions Lesson 1: Introduction
Zoom Grid Slides The Zoom Grid
Dilations in the Coordinate Plane
Mapping Rule: (x, y) → (kx, ky)
The Smartphone Effect
When you "pinch-to-zoom" on a map, the phone's GPU is performing a dilation at 60 frames per second.
Under the Hood:
1. The origin is usually the center of the screen.
2. Every pixel coordinate \((x, y)\) is multiplied by a zoom factor \(k\).
The Coordinate Rule
Dilating about the Origin (0,0)
\((x, y) \rightarrow (kx, ky)\)
Input
Point \(P(4, -6)\)
Output (\(k=2\))
Point \(P'(8, -12)\)
Fractional Scaling
When \(k\) is a fraction (e.g., \(k = \frac{1}{2}\) or \(k = 0.5\)), the coordinates move half-way toward the origin.
// Reduction with k = 1/2
A(10, 20) → A'(5, 10)
B(-4, 0) → B'(-2, 0)
Warning: Off-Center!
If the center of dilation is not the origin, the \((kx, ky)\) rule will not work.
Instead, we scale the vector (the distance) from the center to each point.
More on this in the next activity!
Coordinate Dilations Practice Coordinate Scaling Practice
Target Rule: (x, y) → (kx, ky)
Name: ______________________ Date: __________
1. Point Transformations
Complete the table by applying the given scale factor \(k\) to the pre-image point. The center of dilation is the origin (0, 0).
Pre-Image Point \(P\) Scale Factor \(k\) Calculation Image Point \(P'\) (2, 5) \(k = 3\) \( (2 \cdot 3, 5 \cdot 3) \) (-8, 4) \(k = \frac{1}{2}\) (0, -10) \(k = 0.2\) (-6, -3) \(k = 1.5\)
2. Graphical Dilation
Task: Quadrilateral \(ABCD\) has vertices:
Dilation Rule: \(k = 2.5\), Center (0, 0)
New Coordinates:
\(A'\):
\(B'\):
\(C'\):
\(D'\):
Origin (0,0) at Center
3. Find the Scale Factor
Triangle \(\triangle XYZ\) has vertex \(X(5, -2)\). After a dilation centered at the origin, the image vertex is \(X'(12.5, -5)\).
What is the scale factor \(k\)?
k = ____
Reflection Question:
If point \(P\) is at \((3, 4)\) and \(k = -2\), where would the image be? (Hint: Think about what multiplying by a negative number does to a coordinate's direction). Is this still a dilation?
Geometry Unit 4 • Lesson 2: Coordinate Transformations
Mapping Similarity Slides Mapping Similarity
Defining the Connection
Are they the same?
A designer creates a logo. They rotate it, flip it, and scale it up for a billboard.
The Big Question:
Is the small business card logo "the same" as the giant billboard logo?
> Translation: Yes, they are SIMILAR.
G
G
Similarity Defined
Two figures are Similar if and only if there exists a similarity transformation that maps one onto the other.
A Similarity Transformation includes:
Rigid Motions
Dilation
Step-by-Step Mapping
STEP 1
Translate
Align a vertex of the pre-image with a vertex of the image.
STEP 2
Dilate
Scale the sides using a scale factor \(k\).
STEP 3
Rotate / Reflect
Orient the figure to perfectly overlap the target image.
~
Similarity Notation
"Tilde" (\(\sim\)) means "is similar to"
\(\triangle ABC \sim \triangle DEF\)
Angles Match. Sides Scale.
Logo Logic Activity Logo Logic
Case Study: Similarity Transformations
Detective Team
The Client Brief
The "Apex" brand uses a stylized triangle logo. To prove their branding is consistent across all media, you must provide a mathematical Similarity Transformation that maps the business card logo (Pre-Image) onto the shop sign logo (Image).
1
Coordinate Mapping
Pre-Image: Business Card
Vertex \(A(0, 0)\), \(B(2, 0)\), \(C(1, 2)\)
Image: Shop Sign
Vertex \(A'(5, 5)\), \(B'(15, 5)\), \(C'(10, 15)\)
Identify the steps:
Step 1: Translation Vector
Step 2: Scale Factor (k)
GRAPH PAPER AREA - VISUALIZE THE SHIFT
2
The Complex Proof
Two polygons are similar if one can be mapped to the other using rigid motions and a dilation. Describe the transformation sequence below.
L
Pre-Image L
L
Target Image L'
1. Describe the Dilation (What is \(k\)?):
2. Describe the Rigid Motion (Rotation/Reflection?):
similarity.transf.03 Unit: Non-Rigid Motions Page 01 of 01
Proportional Proofs Slides Proportional Proofs
Angles, Sides, and Similarity Criteria
AA SAS SSS
The Forensic Eye
Detectives often have photos with a ruler or common object placed in the shot.
// Logic Path:
If we know the real-life height of the ruler, we can create a Similarity Ratio to find the height of anything else in the photo.
REFERENCE RULER
What Changes? What Stays?
Invariant (Same)
Angle Measures
Parallelism of Lines
Orientation
Variant (Different)
Side Lengths
Perimeter
Area (Scales by \(k^2\))
AA Similarity Criterion
If two angles of one triangle are congruent to two angles of another...
Then the triangles MUST be similar.
"Why? Because the third angle is automatically fixed by the 180° rule, and dilations preserve angles."
60° 40°
Sides and Ratios
SSS Similarity
All three corresponding sides must have the same ratio.
a/d = b/e = c/f = k
SAS Similarity
Two sides are proportional, and the included angle is congruent.
∠A ≅ ∠D AND a/d = b/e
Forensic Similarity Lab Forensic Similarity Lab
Investigation: Crime Scene Reconstruction
Evidence Batch #802-G
Analyst:
Date:
Crime Scene Photo: Photo-A
ENLARGED FOR ANALYSIS
TIMESTAMP: 23:14:02
Note: In the original photo, the reference ruler measures 2 cm tall. The suspect's height in the photo is 7.5 cm tall.
Lab Data: Known Reference
The object on the left is a standard forensic ruler. Its real-world height is 12 inches (1 foot) .
1. Calculate the Scale Ratio (\(k\))
\( \frac{\text{Photo Ruler}}{\text{Real Ruler}} \)
2. Calculate Real Suspect Height
\( \text{Suspect Height (Photo)} \div k \)
Case Supplement: Triangle Triage
Analyze each pair of triangles. Determine if they are similar. If yes, state the criterion (AA, SAS, SSS) and the scale factor.
PAIR A SAS INVESTIGATION
4
3
2
1.5
Similar? ____
Criterion: ____
PAIR B AA INVESTIGATION
70°
70°
70°
40°
Similar? ____
Criterion: ____
CRITICAL ANALYSIS: Why does AA work?
If two triangles have two congruent angles, we know their shape is preserved. How does the Triangle Sum Theorem (180°) prove that the third angle is also congruent? Write your explanation below.
case-file: similarity-properties-v4 geolab-reconstruction-dept
Blueprint Mastery Slides Blueprint Mastery
Advanced Similarity & Modeling
The Architect's Challenge
A blueprint is a scaled copy of a building.
// Scale 1:50
"If the living room on the blueprint is 4 inches wide, how wide is the real room?"
Dilation Center: Reference Point (0,0)
PLAN_REVISION_V2.0
Locating the Center
To find the Center of Dilation between two similar figures:
The Method:
Connect corresponding vertices with lines.
The point where all lines intersect is the center.
The Square Principle
\(k \rightarrow k^2\)
When you double the sides of a square (\(k=2\))...
The Area becomes 4 times larger!
\[\text{New Area} = k^2 \cdot \text{Old Area}\]
Pre-Image (1x1)
THE ARCHITECT'S ESCAPE
Final Assessment Challenge
"To escape the room, you must solve a series of similarity puzzles to find the missing coordinates of the blueprint's master key."
4 Puzzles 1 Key 60 Minutes
Architect Escape Room Activity The Architect's Escape
Session Security Clearance
TEAM: ________
TIME: ____
A master architect has hidden the exit code within a series of geometric blueprints. Use your knowledge of dilations, similarity transformations, and scale factors to unlock each stage. The final exit code is the sum of all your numerical answers.
1
The Coordinate Key
The main vault door is represented by point \(V(4, -6)\). To unlock it, dilate the point about the origin with a scale factor of \(k = 1.5\).
Resulting Coordinates \(V'\):
(______, ______)
2
The Proportional Plan
Two rooms on the blueprint are similar triangles. Find the missing dimension \(x\) to reveal the second part of the code.
12 ft 9 ft
x 6 ft
X = ______
3
The Area Expansion
A server room on the blueprint has an area of 20 square inches. The master plan expands the server room using a scale factor of k = 4.
Question:
What is the Area of the New Server Room?
______
Remember: Area scales by the square of the scale factor!
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