Dilation Discovery Slides Geometry Series
Dilation
Discovery
Expanding our perspective on the coordinate plane
The Projector Effect
When you move a projector further from the wall, the image gets bigger.
Critical Question:
"Does the image actually change shape, or just change size?"
Small Image
Dilated Image
01. What is a Dilation?
1
Non-Rigid Transformation
Unlike rotations or reflections, dilations change the size of the figure.
2
Preservation
Angle measures remain exactly the same. Orientation stays the same.
3
Proportionality
All side lengths change by the same ratio (the Scale Factor).
The Golden Rule
Similarity ≠ Congruence
Dilated figures are SIMILAR because their shapes are identical, but their sizes differ.
The Scale Factor (k)
Expansion
k > 1
The image gets larger than the pre-image.
Example: k = 2
Reduction
0 < k < 1
The image gets smaller than the pre-image.
Example: k = 0.5
Ratio Formula:
k = \frac{\text{Image Length}}{\text{Pre-Image Length}}
Coordinate Rules
Crucial Note
When the center of dilation is the origin (0, 0):
(x, y) \rightarrow (kx, ky)
Just multiply every coordinate by k!
Original
(2, 4)
Dilated (k=3)
(6, 12)
Quick Check
If a triangle with vertices at (1, 1), (3, 1), and (1, 4) is dilated by a scale factor of k = 2 from the origin...
What are the new coordinates?
Angles stay the same? YES
Sides get 2x longer? YES
Same shape? YES
Are they congruent? NO
Scale Factor Shift Worksheet Scale Factor Shift
Lesson 01: Dilations on the Coordinate Plane
Name:
Date:
k
Scale Factor
k = \frac{\text{Image}}{\text{Pre-Image}}
Rule
Notation
(x, y) \rightarrow (kx, ky)
Type
Category
Similarity Transformation
1 Finding the Ratio
Determine the scale factor (k) for each dilation shown below. State whether it is an Expansion or a Reduction .
Pre-image Image
k =
Type:
Pre-image: 8 units Image: 2 units
k =
Type:
2 Mapping the Shift
3. Square $ABCD$ has vertices $A(0, 0)$, $B(4, 0)$, $C(4, 4)$, and $D(0, 4)$. Dilate the square about the origin using a scale factor of k = 1.5.
Pre-Image Coordinates
$A(0, 0) \rightarrow A'($ _______ , _______ $)$
$B(4, 0) \rightarrow B'($ _______ , _______ $)$
$C(4, 4) \rightarrow C'($ _______ , _______ $)$
$D(0, 4) \rightarrow D'($ _______ , _______ $)$
GRAPHER SPACE
4. A triangle is dilated by k = 1/4. If the image point is $P'(-2, 5)$, what were the coordinates of the original pre-image point $P$?
Challenge: The Spotlight
A small flashlight project is 12 inches from a wall. It shines through a stencil that is 2 inches tall. On the wall, the projected image is 10 inches tall.
A. Calculate the Scale Factor (k)
B. If the stencil's width is 3 in, what is the image width?
Dilation Lab Teacher Guide Dilation Lab Facilitator Guide
Topic: Dilations & Scale Factors | 10th Grade Geometry
Pacing
55 Minutes
Learning Objective
Students will be able to perform and describe dilations on the coordinate plane and identify the scale factor $k$ by comparing image and pre-image lengths. They will distinguish between expansion ($k > 1$) and reduction ($0 < k < 1$).
Materials
Projector or High-intensity flashlight
Cardboard cutouts (Triangles/Rectangles)
Graph Paper & Rulers
"Scale Factor Shift" Worksheet
01
The Hook: Shadows & Size (10 min)
Place a cardboard triangle between a light source and the wall. Slowly move the triangle toward the light, then toward the wall. Ask students to observe the "Center of Dilation" (the light source).
Key Prompt:
"As the shadow gets larger, does the triangle's shape change? Do the angles look different, or just the sides?"
02
Notation & Rules (15 min)
Transition to the coordinate plane. Introduce the rule $(x, y) \rightarrow (kx, ky)$. Emphasize that for the purpose of this lesson, the Origin (0,0) is the center of dilation.
Common Misconception
Students often add $k$ instead of multiplying. Remind them: Dilation is about multiplication (scaling), not shifting.
Scale Factor check
If $k = 1$, the figure stays identical. If $k = -1$, it's a 180° rotation! (Save for advanced discussion).
03
Dilation Practice (20 min)
Hand out the "Scale Factor Shift" worksheet. Walk around and monitor Problem 3, where students must map coordinates. Ensure they are using 1.5 as the multiplier.
Quick Answer Key
Q1 k = 2.5
Q2 k = 0.25 (or 1/4)
Q3 (A') A'(0, 0)
Q4 Multiply P' by 4:
Similarity Showdown Slides Geometry Proving Ground
Similarity
Showdown
Mastering AA, SAS, and SSS Similarity Theorems
The Critical Difference
Congruent (\cong) Identical Twins
Same Shape
Same Size
Side Ratio = 1:1
Similar (\sim) Scale Models
Same Shape
Different Size
Proportional Sides
AA
Angle-Angle Similarity
If two angles of one triangle are congruent to two angles of another...
The triangles are similar.
SAS
Side-Angle-Side Similarity
One angle is congruent, AND the sides including those angles are proportional.
\angle A \cong \angle D
\frac{AB}{DE} = \frac{AC}{DF}
SSS
Side-Side-Side Similarity
If the corresponding side lengths of two triangles are proportional...
\frac{a}{d} = \frac{b}{e} = \frac{c}{f}
3
:
6
Ratio = 1:2
Ready to Prove It?
Step 1
Identify corresponding parts.
Step 2
Test the ratios or angle congruence.
Step 3
State the theorem (AA, SAS, or SSS).
Tri Similarity Proofs Worksheet Tri-Similarity Proofs
Lesson 02 | Formal Verification Practice
Student:
Score:
A
Detection Phase
Determine if the following triangle pairs are similar. If yes, state the Similarity Theorem (AA, SAS, or SSS) and write the Similarity Statement .
$A$ $B$ $C$ 35° 65° $D$ $E$ $F$ 35° 65°
Similar?
Yes
No
Theorem: _________________
$\Delta ABC \sim \Delta$ _________
4 6 6 9
Similar?
Yes
No
Theorem: _________________
Statement: _______________
B
Verification Lab (Proofs)
Problem 3: The Bow-Tie Medium
Given: $\overline{AB} \parallel \overline{CD}$. Prove: $\Delta ABE \sim \Delta DCE$.
Statements Reasons 1. $\overline{AB} \parallel \overline{CD}$ 1. Given 2. 2. Alt. Interior Angles Theorem 3. $\angle AEB \cong \angle DEC$ 3. 4. 4.
$A$ $B$ $D$ $C$ $E$
Problem 4: Proportional Investigation
If $\Delta LMN$ has sides of 3, 4, 5 and $\Delta PQR$ has sides of 9, 12, 15, use a flow-chart style or written paragraph to prove similarity.
Tri Similarity Key Teacher Guide Answer Key & Teacher Guide
Material: Tri-Similarity Proofs Worksheet
TEACHER USE ONLY
Part A: Detection Phase
Problem 1 Solution:
Similar? Yes
Theorem: AA Similarity
Statement: $\Delta ABC \sim \Delta FDE$ (Order matters! Match 35° to 35° and 65° to 65°)
Problem 2 Solution:
Similar? Yes
Theorem: SAS Similarity
Work: Ratio 1: $\frac{6}{9} = \frac{2}{3}$. Ratio 2: $\frac{4}{6} = \frac{2}{3}$. Included angles are right angles (90°).
Part B: Proof Breakdown
Problem 3: The Bow-Tie (2-Column Proof)
Statements Reasons 1. $\overline{AB} \parallel \overline{CD}$ 1. Given 2. $\angle A \cong \angle D$ or $\angle B \cong \angle C$ 2. Alt. Interior Angles Theorem 3. $\angle AEB \cong \angle DEC$ 3. Vertical Angles Theorem 4. $\Delta ABE \sim \Delta DCE$ 4. AA Similarity Theorem
Pedagogical Note on Problem 4:
"Students should explicitly calculate all three ratios: 3/9, 4/12, and 5/15. All reduce to 1/3. Look for the concluding statement: 'Since all three corresponding side ratios are equal, the triangles are similar by SSS Similarity.'"
Grading Checklist
[ ] Correct Similarity Theorem used?
[ ] Ratios calculated & simplified?
[ ] Statement vertices in correct order?
[ ] Reasons correctly cite properties?
Parallel Proportions Slides Geometry Lesson 03
Parallel
Proportions
Exploring the Side-Splitter Theorem & Midsegments
The Side-Splitter Theorem
If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.
\frac{AD}{DB} = \frac{AE}{EC}
$A$ $B$ $C$ $D$ $E$
The Ratio Relationship
Option 1
Upper : Lower
Compare the segments created by the cut on each side.
Option 2
Side : Side
Compare the upper segment to the entire side length.
Option 3
Side : Bottom
Compare small triangle sides to large triangle sides.
The Midsegment Special
Special Case
A midsegment connects the midpoints of two sides.
It is parallel to the 3rd side.
It is exactly HALF the length of the 3rd side.
Midsegment = \frac{1}{2}(\text{Base})
Construction Trick
How do you divide a board into 5 equal parts without using a ruler?
Hint: It involves parallel lines and the Triangle Proportionality Theorem. We'll solve this in our workshop today!
Side Splitter Practice Worksheet Side Splitter Practice
Lesson 03 | Triangle Proportionality & Midsegments
Ref ID: GEO-03-WS
01
Proportional Segments
6 9 8 x
Find $x$:
x + 2 x + 10 10 15
Solve for $x$:
02
The Midsegment Property
2x - 5 22
Problem 3: In the triangle above, the blue segment is a midsegment.
Problem 4: Conceptual Proof
If a midsegment is parallel to the base, prove that the smaller triangle created is similar to the larger triangle. List the specific similarity theorem used and identify the scale factor.
Construction Trick Teacher Guide The 5-Part Plank Challenge
Teacher Guide: Geometry Workshop Hook
Applied Math
THE MISSION
"You have a wooden plank that is exactly 17.63 inches long. You need to cut it into 5 perfectly equal pieces . You do NOT have a calculator, and your ruler only shows inches/cm (making 17.63/5 nearly impossible to measure precisely)."
The Mathematical Secret
This trick relies on the Triangle Proportionality Theorem . By creating a triangle where one side is easy to divide by 5, we can project those equal divisions onto our "difficult" plank using parallel lines.
Steps for Students:
1 Lay the plank on a large sheet of paper. Draw a line along its edge.
2 From one end of the plank, draw a second line at any angle (about 30° works best).
3 On this new line, mark 5 equal segments of ANY easy length (e.g., 2 inches each).
4 Connect the 5th mark to the end of your plank to close the triangle.
5 Draw lines parallel to that last connection from each of the other marks back to the plank.
Visual Diagram
17.63" PLANK
"Because the lines are parallel and the side segments are equal, the Triangle Proportionality Theorem guarantees the segments on the plank are also equal!"
Facilitation Notes
Why this works:
Each parallel line creates a series of similar triangles that all share the same vertex. The scale factor for each triangle is 1/5, 2/5, 3/5, 4/5, and 1.
Real World Connection:
This is a classic "Carpenter's Trick." It shows that geometry isn't just about solving for $x$—it's about creating systems of measurement.
Altitude Alpha Slides Geometry Masterclass
Altitude
Alpha
The Geometric Mean in Right Triangles
One Triangle, Three Identities
When you draw the altitude to the hypotenuse of a right triangle, it creates two new triangles .
All three triangles are SIMILAR to each other.
$A$ $B$ $C$ $D$
What is Geometric Mean?
The Math
For any two positive numbers a and b, the geometric mean x is the number where:
\frac{a}{x} = \frac{x}{b}
x = \sqrt{ab}
"It's the middle term in a proportional sequence."
Geometric Mean (Altitude)
The altitude is the geometric mean between the two segments of the hypotenuse.
h^2 = x \cdot y
$h$ $x$ $y$
Altitude splits the hypotenuse into $x$ and $y$.
Geometric Mean (Legs)
Theorem:
Each leg is the geometric mean between the hypotenuse and the adjacent segment.
Leg^2 = Adjacent \cdot Whole
"Small piece times big piece equals the leg squared."
Can you solve it?
$h$ 4 9
What is the value of h?
h^2 = 4 \cdot 9
h = ?
Geometric Mean Grid Worksheet Geometric Mean Grid
Lesson 04 | Altitudes & Right Triangles
Sheet No: ALT-04-B
Name:
Geometric Mean
x = \sqrt{a \cdot b}
Altitude Rule
h^2 = (seg_1)(seg_2)
Leg Rule
Leg^2 = (adj)(whole)
01
The Altitude Theorem
$x$ 2 8
Show Your Work
Final Answer: $x =$ _______
12 9 $y$
Show Your Work
Final Answer: $y =$ _______
02
The Leg Theorem
$a$ 3 Whole = 12
Show Your Work
Final Answer: $a =$ _______
10 $z$ Whole = 20
Show Your Work
Final Answer: $z =$ _______
Altitude Alpha Key Teacher Guide Altitude Alpha Key Teacher Guide Master Solution Key
Material: Geometric Mean Grid Worksheet
TEACHER USE ONLY
Part 1: Altitude Theorem Solutions
Problem 1: Solve for $x$
Formula: $h^2 = (seg1) \cdot (seg2)$
Problem 2: Solve for $y$
Formula: $h^2 = (seg1) \cdot (seg2)$
Part 2: Leg Theorem Solutions
Problem 3: Solve for $a$
Formula: $Leg^2 = (adj) \cdot (whole)$
Problem 4: Solve for $z$
Formula: $Leg^2 = (adj) \cdot (whole)$
Common Misconceptions to Monitor
Swapping Formulas: Students often try to use the Altitude rule for Leg problems and vice-versa. Remind them: "Altitude uses the pieces. Leg uses a piece and the WHOLE."Forgetting to Square Root: Many students will stop at $x^2 = 16$. Explicitly check their final step. Shadow Scouter Slides Final Mission
Shadow
Scouter
Calculating Heights with Indirect Measurement
The Challenge
"How do you measure a 40-foot flagpole if you don't have a 40-foot ladder?"
The Sun
Uses shadows as proportions.
The Mirror
Uses reflection angles.
The Proof
Verifies why it works.
Method 01: Shadow Math
AA Similarity
At the same time of day, the sun's rays are parallel, creating similar triangles.
\frac{\text{Object Height}}{\text{Shadow Length}} = \frac{\text{Human Height}}{\text{Shadow Length}}
Method 02: The Mirror
Angle of Incidence = Angle of Reflection
How to do it:
Place a mirror flat on the ground.
Back away until you can see the top of the object in the mirror.
Measure distances to the mirror.
Field Work Prep
Equipment Needed
Long Tape Measure
Laser pointer (optional)
Data Collection Sheet
Camera (to document)
The Rubric
1. Measurement Accuracy (within 10%)
2. Visual Diagram of Triangles
3. Geometric Similarity Proof
4. Calculated Height & Error Analysis
Shadow Scouter Project Guide Shadow Scouter Project
Culminating Field Activity | Geometry Unit 05
The Objective
Apply your knowledge of AA Similarity to measure a large outdoor object (flagpole, tree, or school wall) that is physically impossible to reach. You will document your data, perform calculations, and provide a formal geometric proof for your findings.
Team Roles
Lead Surveyor
Data Logger
Measurement Tech
01
Data Collection Phase
Method A: Shadow Cast
Human Height (h1)
inches
Human Shadow (s1)
inches
Object Shadow (s2)
inches
Method B: Mirror Reflection
Eye Height (e)
inches
Distance to Mirror (d1)
inches
Mirror to Object (d2)
inches
02
Calculation & Verification
A. Proportional Setup
\frac{?}{?} = \frac{?}{?}
B. Calculated Height
______ ft
C. Formal Similarity Proof
Explain WHY these triangles are similar. Identify the theorem used (AA, SAS, SSS) and describe the physical properties that justify the theorem.
Flagpole Finale Rubric Flagpole Finale Rubric
Evaluation Criteria | Unit 05 Culminating Task
Max Points
100
Criteria Performance Level Points Data Accuracy All measurements (heights and shadows/distances) are recorded precisely with units. Measurements are consistent across trials. / 25 Geometric Diagram Diagram correctly illustrates the right triangles formed. Vertices are labeled and right angles are indicated. / 20 Formal Proof Student correctly identifies AA Similarity. Proof explains how parallel sun rays or congruent reflection angles create similarity. / 25 Calculation Logic Ratios are set up correctly. Algebraic steps are shown. Height is converted to appropriate final units (ft/in). / 20 Analysis Student identifies at least two potential sources of error (e.g., ground levelness, rounding) and their impact. / 10
Instructor Evaluation Notes
Total Score:
Letter Grade:
MASTER
PROVED
SIMILAR
