Dilations - Tenth Grade Geometry
Introduction to Dilations
Dilation: A transformation that changes the size of a figure but not its shape
Scale Factor (k): The ratio by which the figure is enlarged or reduced
Center of Dilation: The fixed point from which all points are scaled
Pre-image: The original figure before dilation
Image: The figure after dilation (denoted with prime notation: A')
Key Property: NOT a congruence transformation (size changes, but shape is preserved)
Scale Factor (k): The ratio by which the figure is enlarged or reduced
Center of Dilation: The fixed point from which all points are scaled
Pre-image: The original figure before dilation
Image: The figure after dilation (denoted with prime notation: A')
Key Property: NOT a congruence transformation (size changes, but shape is preserved)
Types of Dilations:
1. Enlargement (k > 1):
• Figure gets bigger
• Scale factor greater than 1
• Example: k = 2 (double the size), k = 3 (triple the size)
2. Reduction (0 < k < 1):
• Figure gets smaller
• Scale factor between 0 and 1
• Example: k = 0.5 (half the size), k = 1/3 (one-third the size)
3. Identity (k = 1):
• Figure stays the same size
• No change occurs
Note: Scale factor cannot be negative or zero in standard dilations
1. Enlargement (k > 1):
• Figure gets bigger
• Scale factor greater than 1
• Example: k = 2 (double the size), k = 3 (triple the size)
2. Reduction (0 < k < 1):
• Figure gets smaller
• Scale factor between 0 and 1
• Example: k = 0.5 (half the size), k = 1/3 (one-third the size)
3. Identity (k = 1):
• Figure stays the same size
• No change occurs
Note: Scale factor cannot be negative or zero in standard dilations
Properties PRESERVED by Dilation:
• Shape
• Angle measures
• Parallelism (parallel lines remain parallel)
• Collinearity (points on a line remain on a line)
• Ratio of lengths within the figure
Properties that CHANGE:
• Size (unless k = 1)
• Distance between points
• Perimeter (multiplied by k)
• Area (multiplied by k²)
• Volume in 3D (multiplied by k³)
• Shape
• Angle measures
• Parallelism (parallel lines remain parallel)
• Collinearity (points on a line remain on a line)
• Ratio of lengths within the figure
Properties that CHANGE:
• Size (unless k = 1)
• Distance between points
• Perimeter (multiplied by k)
• Area (multiplied by k²)
• Volume in 3D (multiplied by k³)
1. Dilations: Graph the Image
Graphing Dilations: Plot the image after applying scale factor
Most Common: Center of dilation at origin (0, 0)
Method: Multiply each coordinate by the scale factor
Most Common: Center of dilation at origin (0, 0)
Method: Multiply each coordinate by the scale factor
Dilation with Center at Origin
Dilation Formula (Center at Origin):
$$D_k(x, y) = (kx, ky)$$
Where:
• $k$ = scale factor
• $(x, y)$ = original coordinates
• $(kx, ky)$ = new coordinates after dilation
Rule: Multiply BOTH x and y coordinates by the SAME scale factor k
$$D_k(x, y) = (kx, ky)$$
Where:
• $k$ = scale factor
• $(x, y)$ = original coordinates
• $(kx, ky)$ = new coordinates after dilation
Rule: Multiply BOTH x and y coordinates by the SAME scale factor k
Example 1: Enlargement (k > 1)
Dilate point A(3, 2) with center at origin and scale factor k = 2
$A(3, 2) \to A'(2 \cdot 3, 2 \cdot 2) = A'(6, 4)$
The point moves twice as far from the origin
Dilate point A(3, 2) with center at origin and scale factor k = 2
$A(3, 2) \to A'(2 \cdot 3, 2 \cdot 2) = A'(6, 4)$
The point moves twice as far from the origin
Example 2: Reduction (0 < k < 1)
Dilate triangle ABC with A(4, 6), B(8, 2), C(2, -4) using scale factor k = 0.5
$A(4, 6) \to A'(0.5 \cdot 4, 0.5 \cdot 6) = A'(2, 3)$
$B(8, 2) \to B'(0.5 \cdot 8, 0.5 \cdot 2) = B'(4, 1)$
$C(2, -4) \to C'(0.5 \cdot 2, 0.5 \cdot (-4)) = C'(1, -2)$
Triangle is reduced to half its original distance from origin
Dilate triangle ABC with A(4, 6), B(8, 2), C(2, -4) using scale factor k = 0.5
$A(4, 6) \to A'(0.5 \cdot 4, 0.5 \cdot 6) = A'(2, 3)$
$B(8, 2) \to B'(0.5 \cdot 8, 0.5 \cdot 2) = B'(4, 1)$
$C(2, -4) \to C'(0.5 \cdot 2, 0.5 \cdot (-4)) = C'(1, -2)$
Triangle is reduced to half its original distance from origin
Example 3: Fractional scale factor
Dilate point P(6, 9) with scale factor k = 1/3
$P(6, 9) \to P'\left(\frac{1}{3} \cdot 6, \frac{1}{3} \cdot 9\right) = P'(2, 3)$
Point is one-third the distance from origin
Dilate point P(6, 9) with scale factor k = 1/3
$P(6, 9) \to P'\left(\frac{1}{3} \cdot 6, \frac{1}{3} \cdot 9\right) = P'(2, 3)$
Point is one-third the distance from origin
Dilation with Center NOT at Origin
Dilation Formula (Center at (a, b)):
$$D_{k,(a,b)}(x, y) = (k(x - a) + a, k(y - b) + b)$$
Where:
• $k$ = scale factor
• $(a, b)$ = center of dilation
• $(x, y)$ = original coordinates
Step-by-Step Process:
1. Translate so center moves to origin: $(x - a, y - b)$
2. Apply dilation: $(k(x - a), k(y - b))$
3. Translate back: $(k(x - a) + a, k(y - b) + b)$
$$D_{k,(a,b)}(x, y) = (k(x - a) + a, k(y - b) + b)$$
Where:
• $k$ = scale factor
• $(a, b)$ = center of dilation
• $(x, y)$ = original coordinates
Step-by-Step Process:
1. Translate so center moves to origin: $(x - a, y - b)$
2. Apply dilation: $(k(x - a), k(y - b))$
3. Translate back: $(k(x - a) + a, k(y - b) + b)$
Example 4: Center not at origin
Dilate point A(5, 7) with center at C(2, 3) and scale factor k = 2
$x' = k(x - a) + a = 2(5 - 2) + 2 = 2(3) + 2 = 8$
$y' = k(y - b) + b = 2(7 - 3) + 3 = 2(4) + 3 = 11$
$A'(8, 11)$
Dilate point A(5, 7) with center at C(2, 3) and scale factor k = 2
$x' = k(x - a) + a = 2(5 - 2) + 2 = 2(3) + 2 = 8$
$y' = k(y - b) + b = 2(7 - 3) + 3 = 2(4) + 3 = 11$
$A'(8, 11)$
2. Dilations: Find the Coordinates
Steps to Find Coordinates After Dilation:
If center is at origin:
Step 1: Identify the scale factor k
Step 2: Multiply each x-coordinate by k
Step 3: Multiply each y-coordinate by k
Step 4: Write new coordinates as (kx, ky)
If center is NOT at origin (a, b):
Step 1: Identify scale factor k and center (a, b)
Step 2: Use formula: $x' = k(x - a) + a$ and $y' = k(y - b) + b$
Step 3: Calculate new coordinates
Step 4: Simplify
If center is at origin:
Step 1: Identify the scale factor k
Step 2: Multiply each x-coordinate by k
Step 3: Multiply each y-coordinate by k
Step 4: Write new coordinates as (kx, ky)
If center is NOT at origin (a, b):
Step 1: Identify scale factor k and center (a, b)
Step 2: Use formula: $x' = k(x - a) + a$ and $y' = k(y - b) + b$
Step 3: Calculate new coordinates
Step 4: Simplify
Example 1: Multiple vertices
Rectangle PQRS has vertices P(1, 2), Q(4, 2), R(4, 5), S(1, 5). Find P'Q'R'S' after dilation with k = 3, center at origin.
$P(1, 2) \to P'(3, 6)$
$Q(4, 2) \to Q'(12, 6)$
$R(4, 5) \to R'(12, 15)$
$S(1, 5) \to S'(3, 15)$
Rectangle PQRS has vertices P(1, 2), Q(4, 2), R(4, 5), S(1, 5). Find P'Q'R'S' after dilation with k = 3, center at origin.
$P(1, 2) \to P'(3, 6)$
$Q(4, 2) \to Q'(12, 6)$
$R(4, 5) \to R'(12, 15)$
$S(1, 5) \to S'(3, 15)$
Example 2: Negative coordinates
Dilate point M(-4, 6) with k = 0.5, center at origin
$M(-4, 6) \to M'(0.5 \cdot (-4), 0.5 \cdot 6) = M'(-2, 3)$
Dilate point M(-4, 6) with k = 0.5, center at origin
$M(-4, 6) \to M'(0.5 \cdot (-4), 0.5 \cdot 6) = M'(-2, 3)$
Example 3: Working backwards
After dilation with k = 4 and center at origin, point T' is at (12, -8). Find original point T.
If $(x, y) \to (12, -8)$ with k = 4:
$4x = 12 \implies x = 3$
$4y = -8 \implies y = -2$
Original point: T(3, -2)
After dilation with k = 4 and center at origin, point T' is at (12, -8). Find original point T.
If $(x, y) \to (12, -8)$ with k = 4:
$4x = 12 \implies x = 3$
$4y = -8 \implies y = -2$
Original point: T(3, -2)
3. Dilations: Find Length, Perimeter, and Area
Key Concept: Dilation affects measurements differently
Linear measurements (length, perimeter): Multiplied by k
Area measurements: Multiplied by k²
Volume measurements (3D): Multiplied by k³
Linear measurements (length, perimeter): Multiplied by k
Area measurements: Multiplied by k²
Volume measurements (3D): Multiplied by k³
Effect of Dilation on Measurements:
1. Length of a side:
$$\text{New length} = k \times \text{Original length}$$
2. Perimeter:
$$\text{New perimeter} = k \times \text{Original perimeter}$$
3. Area:
$$\text{New area} = k^2 \times \text{Original area}$$
4. Volume (for 3D shapes):
$$\text{New volume} = k^3 \times \text{Original volume}$$
1. Length of a side:
$$\text{New length} = k \times \text{Original length}$$
2. Perimeter:
$$\text{New perimeter} = k \times \text{Original perimeter}$$
3. Area:
$$\text{New area} = k^2 \times \text{Original area}$$
4. Volume (for 3D shapes):
$$\text{New volume} = k^3 \times \text{Original volume}$$
Example 1: Length
A segment AB has length 5 cm. After dilation with k = 3, what is the length of A'B'?
New length = $k \times$ Original length
New length = $3 \times 5 = 15$ cm
Answer: A'B' = 15 cm
A segment AB has length 5 cm. After dilation with k = 3, what is the length of A'B'?
New length = $k \times$ Original length
New length = $3 \times 5 = 15$ cm
Answer: A'B' = 15 cm
Example 2: Perimeter
Triangle has perimeter of 24 cm. After dilation with k = 2.5, what is new perimeter?
New perimeter = $k \times$ Original perimeter
New perimeter = $2.5 \times 24 = 60$ cm
Answer: New perimeter = 60 cm
Triangle has perimeter of 24 cm. After dilation with k = 2.5, what is new perimeter?
New perimeter = $k \times$ Original perimeter
New perimeter = $2.5 \times 24 = 60$ cm
Answer: New perimeter = 60 cm
Example 3: Area
Rectangle has area 20 square units. After dilation with k = 3, what is new area?
New area = $k^2 \times$ Original area
New area = $3^2 \times 20 = 9 \times 20 = 180$ square units
Answer: New area = 180 square units
Rectangle has area 20 square units. After dilation with k = 3, what is new area?
New area = $k^2 \times$ Original area
New area = $3^2 \times 20 = 9 \times 20 = 180$ square units
Answer: New area = 180 square units
Example 4: Reduction effect
Square has area 100 cm². After dilation with k = 0.5, what is new area?
New area = $k^2 \times$ Original area
New area = $(0.5)^2 \times 100 = 0.25 \times 100 = 25$ cm²
Answer: New area = 25 cm² (one-fourth of original)
Square has area 100 cm². After dilation with k = 0.5, what is new area?
New area = $k^2 \times$ Original area
New area = $(0.5)^2 \times 100 = 0.25 \times 100 = 25$ cm²
Answer: New area = 25 cm² (one-fourth of original)
Important Pattern:
If scale factor is k = 2:
• Lengths are multiplied by 2
• Perimeter is multiplied by 2
• Area is multiplied by 2² = 4
• Volume would be multiplied by 2³ = 8
This is why doubling dimensions doesn't double area - it quadruples it!
If scale factor is k = 2:
• Lengths are multiplied by 2
• Perimeter is multiplied by 2
• Area is multiplied by 2² = 4
• Volume would be multiplied by 2³ = 8
This is why doubling dimensions doesn't double area - it quadruples it!
4. Dilations: Find the Scale Factor
Finding Scale Factor: Determine k when given pre-image and image
Methods: Compare coordinates, lengths, or use the ratio formula
Key Property: Scale factor is constant for all corresponding parts
Methods: Compare coordinates, lengths, or use the ratio formula
Key Property: Scale factor is constant for all corresponding parts
Scale Factor Formulas:
Method 1: Using Coordinates (center at origin)
$$k = \frac{x'}{x} = \frac{y'}{y}$$
Method 2: Using Lengths
$$k = \frac{\text{Image length}}{\text{Pre-image length}}$$
Method 3: Using Area
$$k = \sqrt{\frac{\text{Image area}}{\text{Pre-image area}}}$$
Method 4: Using Perimeter
$$k = \frac{\text{Image perimeter}}{\text{Pre-image perimeter}}$$
Method 1: Using Coordinates (center at origin)
$$k = \frac{x'}{x} = \frac{y'}{y}$$
Method 2: Using Lengths
$$k = \frac{\text{Image length}}{\text{Pre-image length}}$$
Method 3: Using Area
$$k = \sqrt{\frac{\text{Image area}}{\text{Pre-image area}}}$$
Method 4: Using Perimeter
$$k = \frac{\text{Image perimeter}}{\text{Pre-image perimeter}}$$
Example 1: From coordinates
Point A(2, 3) is dilated to A'(6, 9). Find the scale factor.
Using x-coordinates: $k = \frac{x'}{x} = \frac{6}{2} = 3$
Verify with y-coordinates: $k = \frac{y'}{y} = \frac{9}{3} = 3$ ✓
Answer: k = 3 (enlargement)
Point A(2, 3) is dilated to A'(6, 9). Find the scale factor.
Using x-coordinates: $k = \frac{x'}{x} = \frac{6}{2} = 3$
Verify with y-coordinates: $k = \frac{y'}{y} = \frac{9}{3} = 3$ ✓
Answer: k = 3 (enlargement)
Example 2: From side lengths
Side AB = 4 cm, after dilation A'B' = 10 cm. Find k.
$k = \frac{\text{A'B'}}{\text{AB}} = \frac{10}{4} = 2.5$
Answer: k = 2.5 (enlargement by factor of 2.5)
Side AB = 4 cm, after dilation A'B' = 10 cm. Find k.
$k = \frac{\text{A'B'}}{\text{AB}} = \frac{10}{4} = 2.5$
Answer: k = 2.5 (enlargement by factor of 2.5)
Example 3: Reduction
Point P(12, 8) dilated to P'(3, 2). Find k.
$k = \frac{x'}{x} = \frac{3}{12} = \frac{1}{4} = 0.25$
Verify: $k = \frac{y'}{y} = \frac{2}{8} = \frac{1}{4}$ ✓
Answer: k = 0.25 or 1/4 (reduction to one-fourth)
Point P(12, 8) dilated to P'(3, 2). Find k.
$k = \frac{x'}{x} = \frac{3}{12} = \frac{1}{4} = 0.25$
Verify: $k = \frac{y'}{y} = \frac{2}{8} = \frac{1}{4}$ ✓
Answer: k = 0.25 or 1/4 (reduction to one-fourth)
Example 4: From area
Triangle has area 15 cm², dilated triangle has area 60 cm². Find k.
$k^2 = \frac{\text{New area}}{\text{Original area}} = \frac{60}{15} = 4$
$k = \sqrt{4} = 2$
Answer: k = 2
Triangle has area 15 cm², dilated triangle has area 60 cm². Find k.
$k^2 = \frac{\text{New area}}{\text{Original area}} = \frac{60}{15} = 4$
$k = \sqrt{4} = 2$
Answer: k = 2
5. Dilations: Find the Scale Factor and Center of Dilation
Center of Dilation: The fixed point from which dilation occurs
Key Property: The center point does not move during dilation
Finding Center: Use corresponding points and scale factor
Key Property: The center point does not move during dilation
Finding Center: Use corresponding points and scale factor
Method to Find Center of Dilation
Steps to Find Center of Dilation:
Step 1: Find the scale factor k using any pair of corresponding points
$$k = \frac{x'}{x} \text{ (if center is at origin)}$$
Step 2: If center is NOT at origin, use the formula:
For points $A(x_1, y_1)$ and $A'(x_1', y_1')$:
$$x_c = \frac{kx_1 - x_1'}{k - 1}$$
$$y_c = \frac{ky_1 - y_1'}{k - 1}$$
Step 3: Verify with another pair of corresponding points
Alternative Method: Draw lines through corresponding points; they intersect at center
Step 1: Find the scale factor k using any pair of corresponding points
$$k = \frac{x'}{x} \text{ (if center is at origin)}$$
Step 2: If center is NOT at origin, use the formula:
For points $A(x_1, y_1)$ and $A'(x_1', y_1')$:
$$x_c = \frac{kx_1 - x_1'}{k - 1}$$
$$y_c = \frac{ky_1 - y_1'}{k - 1}$$
Step 3: Verify with another pair of corresponding points
Alternative Method: Draw lines through corresponding points; they intersect at center
Example 1: Find center at origin
Point B(3, 4) dilated to B'(9, 12). Where is the center?
Find scale factor: $k = \frac{9}{3} = 3$ and $\frac{12}{4} = 3$ ✓
Check if origin is center: $(3, 4) \to (3 \cdot 3, 3 \cdot 4) = (9, 12)$ ✓
Answer: Center is at origin (0, 0), k = 3
Point B(3, 4) dilated to B'(9, 12). Where is the center?
Find scale factor: $k = \frac{9}{3} = 3$ and $\frac{12}{4} = 3$ ✓
Check if origin is center: $(3, 4) \to (3 \cdot 3, 3 \cdot 4) = (9, 12)$ ✓
Answer: Center is at origin (0, 0), k = 3
Example 2: Find center NOT at origin
Point C(4, 2) dilated to C'(7, 5) with k = 2. Find center.
Using formulas:
$x_c = \frac{kx_1 - x_1'}{k - 1} = \frac{2(4) - 7}{2 - 1} = \frac{8 - 7}{1} = 1$
$y_c = \frac{ky_1 - y_1'}{k - 1} = \frac{2(2) - 5}{2 - 1} = \frac{4 - 5}{1} = -1$
Answer: Center of dilation is (1, -1), k = 2
Verify: $(4, 2)$ with center $(1, -1)$ and $k = 2$:
$x' = 2(4 - 1) + 1 = 2(3) + 1 = 7$ ✓
$y' = 2(2 - (-1)) + (-1) = 2(3) - 1 = 5$ ✓
Point C(4, 2) dilated to C'(7, 5) with k = 2. Find center.
Using formulas:
$x_c = \frac{kx_1 - x_1'}{k - 1} = \frac{2(4) - 7}{2 - 1} = \frac{8 - 7}{1} = 1$
$y_c = \frac{ky_1 - y_1'}{k - 1} = \frac{2(2) - 5}{2 - 1} = \frac{4 - 5}{1} = -1$
Answer: Center of dilation is (1, -1), k = 2
Verify: $(4, 2)$ with center $(1, -1)$ and $k = 2$:
$x' = 2(4 - 1) + 1 = 2(3) + 1 = 7$ ✓
$y' = 2(2 - (-1)) + (-1) = 2(3) - 1 = 5$ ✓
Example 3: Find both k and center
Point D(6, 4) dilated to D'(3, 2). Find k and center.
Step 1: Find k
$k = \frac{3}{6} = 0.5$ and $\frac{2}{4} = 0.5$ ✓
Step 2: Check if center is at origin
$(6, 4) \to (0.5 \cdot 6, 0.5 \cdot 4) = (3, 2)$ ✓
Answer: k = 0.5, center at origin (0, 0)
Point D(6, 4) dilated to D'(3, 2). Find k and center.
Step 1: Find k
$k = \frac{3}{6} = 0.5$ and $\frac{2}{4} = 0.5$ ✓
Step 2: Check if center is at origin
$(6, 4) \to (0.5 \cdot 6, 0.5 \cdot 4) = (3, 2)$ ✓
Answer: k = 0.5, center at origin (0, 0)
6. Dilations and Parallel Lines
Key Property: Dilations preserve parallelism
Meaning: If two lines are parallel before dilation, they remain parallel after dilation
Why: Dilation preserves angles, including the angles that make lines parallel
Meaning: If two lines are parallel before dilation, they remain parallel after dilation
Why: Dilation preserves angles, including the angles that make lines parallel
Properties Preserved by Dilations:
1. Parallelism:
• Parallel lines remain parallel
• Corresponding angles stay equal
• Alternate interior angles stay equal
2. Angle Measures:
• All angles keep the same measure
• Shape is preserved (similar figures)
3. Collinearity:
• Points on a line remain on a line
• The line may move but stays straight
4. Ratios:
• Ratios of lengths within figure stay the same
• Example: If AB:BC = 2:3, then A'B':B'C' = 2:3
What Changes:
• Distance between parallel lines (multiplied by k)
• Actual lengths (multiplied by k)
• Areas (multiplied by k²)
1. Parallelism:
• Parallel lines remain parallel
• Corresponding angles stay equal
• Alternate interior angles stay equal
2. Angle Measures:
• All angles keep the same measure
• Shape is preserved (similar figures)
3. Collinearity:
• Points on a line remain on a line
• The line may move but stays straight
4. Ratios:
• Ratios of lengths within figure stay the same
• Example: If AB:BC = 2:3, then A'B':B'C' = 2:3
What Changes:
• Distance between parallel lines (multiplied by k)
• Actual lengths (multiplied by k)
• Areas (multiplied by k²)
Example 1: Parallel sides of rectangle
Rectangle ABCD has parallel sides AB ∥ CD. After dilation with k = 2, are A'B' and C'D' still parallel?
Answer: YES
Dilations preserve parallelism, so A'B' ∥ C'D'
Additionally:
• If AB = 5, then A'B' = 10
• If CD = 5, then C'D' = 10
• Both sides are still equal and parallel
Rectangle ABCD has parallel sides AB ∥ CD. After dilation with k = 2, are A'B' and C'D' still parallel?
Answer: YES
Dilations preserve parallelism, so A'B' ∥ C'D'
Additionally:
• If AB = 5, then A'B' = 10
• If CD = 5, then C'D' = 10
• Both sides are still equal and parallel
Example 2: Distance between parallel lines
Two parallel lines are 3 units apart. After dilation with k = 4, how far apart are the image lines?
Distance between parallel lines is multiplied by k:
New distance = $4 \times 3 = 12$ units
Answer: 12 units apart (still parallel)
Two parallel lines are 3 units apart. After dilation with k = 4, how far apart are the image lines?
Distance between parallel lines is multiplied by k:
New distance = $4 \times 3 = 12$ units
Answer: 12 units apart (still parallel)
Example 3: Trapezoid
Trapezoid has parallel bases of 6 cm and 10 cm, height 4 cm. After dilation with k = 1.5, find new dimensions.
New base 1: $1.5 \times 6 = 9$ cm
New base 2: $1.5 \times 10 = 15$ cm
New height: $1.5 \times 4 = 6$ cm
The bases remain parallel, just scaled proportionally
Trapezoid has parallel bases of 6 cm and 10 cm, height 4 cm. After dilation with k = 1.5, find new dimensions.
New base 1: $1.5 \times 6 = 9$ cm
New base 2: $1.5 \times 10 = 15$ cm
New height: $1.5 \times 4 = 6$ cm
The bases remain parallel, just scaled proportionally
Why Dilations Preserve Parallelism:
Parallel lines have the same slope. Under dilation:
• Line through points $(x_1, y_1)$ and $(x_2, y_2)$ has slope $m = \frac{y_2 - y_1}{x_2 - x_1}$
• After dilation: points become $(kx_1, ky_1)$ and $(kx_2, ky_2)$
• New slope: $m' = \frac{ky_2 - ky_1}{kx_2 - kx_1} = \frac{k(y_2 - y_1)}{k(x_2 - x_1)} = \frac{y_2 - y_1}{x_2 - x_1} = m$
Slope doesn't change, so parallelism is preserved!
Parallel lines have the same slope. Under dilation:
• Line through points $(x_1, y_1)$ and $(x_2, y_2)$ has slope $m = \frac{y_2 - y_1}{x_2 - x_1}$
• After dilation: points become $(kx_1, ky_1)$ and $(kx_2, ky_2)$
• New slope: $m' = \frac{ky_2 - ky_1}{kx_2 - kx_1} = \frac{k(y_2 - y_1)}{k(x_2 - x_1)} = \frac{y_2 - y_1}{x_2 - x_1} = m$
Slope doesn't change, so parallelism is preserved!
Dilation Formulas Quick Reference
| Center Location | Formula | Example |
|---|---|---|
| Origin (0, 0) | $(x, y) \to (kx, ky)$ | k = 2: $(3, 4) \to (6, 8)$ |
| Point (a, b) | $(x, y) \to (k(x-a)+a, k(y-b)+b)$ | k = 2, center (1,1): $(3, 4) \to (5, 7)$ |
Effect on Measurements
| Measurement | Effect of Scale Factor k | Example (k = 2) |
|---|---|---|
| Side Length | Multiplied by k | 5 cm → 10 cm |
| Perimeter | Multiplied by k | 20 cm → 40 cm |
| Area | Multiplied by k² | 25 cm² → 100 cm² |
| Volume (3D) | Multiplied by k³ | 8 cm³ → 64 cm³ |
| Angle Measure | No change | 60° → 60° |
Scale Factor Types
| Scale Factor k | Type | Effect | Example |
|---|---|---|---|
| k > 1 | Enlargement | Figure gets bigger | k = 2, k = 3, k = 1.5 |
| k = 1 | Identity | No change | Figure stays same |
| 0 < k < 1 | Reduction | Figure gets smaller | k = 0.5, k = 1/3, k = 0.25 |
Properties Comparison
| Property | Preserved? | Notes |
|---|---|---|
| Size | NO | Changes by factor k (unless k = 1) |
| Shape | YES | Pre-image and image are similar |
| Angle measures | YES | All angles stay the same |
| Parallelism | YES | Parallel lines remain parallel |
| Collinearity | YES | Points on line stay on line |
| Distance ratios | YES | Ratios within figure preserved |
| Actual distances | NO | Multiplied by k |
| Orientation | YES | Figure faces same direction |
Key Formulas Summary
| To Find | Formula | Use When |
|---|---|---|
| Image coordinates | $(kx, ky)$ | Center at origin |
| Scale factor from coordinates | $k = \frac{x'}{x} = \frac{y'}{y}$ | Center at origin |
| Scale factor from lengths | $k = \frac{\text{Image length}}{\text{Pre-image length}}$ | Any dilation |
| Scale factor from area | $k = \sqrt{\frac{\text{Image area}}{\text{Pre-image area}}}$ | Area given |
| New perimeter | $P' = k \cdot P$ | Any dilation |
| New area | $A' = k^2 \cdot A$ | Any dilation |
| Center of dilation | $x_c = \frac{kx - x'}{k-1}$, $y_c = \frac{ky - y'}{k-1}$ | When k and points known |
Success Tips for Dilations:
✓ Dilation changes SIZE but preserves SHAPE (similarity transformation)
✓ Center at origin: $(x, y) \to (kx, ky)$ - multiply both coordinates by k
✓ k > 1: enlargement; 0 < k < 1: reduction; k = 1: no change
✓ Perimeter is multiplied by k; Area is multiplied by k²
✓ Scale factor = $\frac{\text{image}}{\text{pre-image}}$ for any corresponding measurement
✓ Dilations preserve: angles, parallelism, shape, ratios
✓ Dilations change: size, distances, perimeter, area
✓ NOT a congruence transformation (unlike translation, reflection, rotation)
✓ Pre-image and image are SIMILAR, not congruent (unless k = 1)
✓ Parallel lines remain parallel after dilation!
✓ Dilation changes SIZE but preserves SHAPE (similarity transformation)
✓ Center at origin: $(x, y) \to (kx, ky)$ - multiply both coordinates by k
✓ k > 1: enlargement; 0 < k < 1: reduction; k = 1: no change
✓ Perimeter is multiplied by k; Area is multiplied by k²
✓ Scale factor = $\frac{\text{image}}{\text{pre-image}}$ for any corresponding measurement
✓ Dilations preserve: angles, parallelism, shape, ratios
✓ Dilations change: size, distances, perimeter, area
✓ NOT a congruence transformation (unlike translation, reflection, rotation)
✓ Pre-image and image are SIMILAR, not congruent (unless k = 1)
✓ Parallel lines remain parallel after dilation!