The Central Idea
A scale factor tells how much every corresponding length in a figure is multiplied when the figure is enlarged or reduced.
If a length \(L\) becomes \(L'\), then
\[ \boxed{k=\frac{L'}{L}}. \]The same factor must apply to every pair of corresponding lengths when shape is preserved.
What Is Similarity?
Two figures are similar when they have the same shape even if they have different sizes.
In similar figures:
- corresponding angles are congruent;
- corresponding lengths are proportional.
Similarity preserves shape while allowing size to change.
Congruence versus Similarity
Congruent figures have the same shape and the same size.
Similar figures have the same shape but may have different sizes.
Congruence can therefore be viewed as similarity with scale factor
\[ \boxed{k=1}. \]Defining the Scale Factor
Suppose a segment of length \(a\) corresponds to a segment of length \(b\) after scaling.
The scale factor from the first figure to the second is
\[ \boxed{k=\frac{b}{a}}. \]The direction matters: reversing the comparison gives the reciprocal scale factor.
Scale Factor Is Directional
If a figure is enlarged from side length \(4\) to side length \(10\), then the scale factor from the original to the image is
\[ \frac{10}{4}=\frac52. \]From the image back to the original, the scale factor is
\[ \frac{4}{10}=\frac25. \]These factors are reciprocals.
Enlargements
If
\[ k>1, \]the image is an enlargement of the original.
Every corresponding length becomes larger by the same multiplicative factor.
Reductions
If
\[ 0Every corresponding length becomes smaller by the same multiplicative factor.
Scale Factor of One
If
\[ k=1, \]corresponding lengths do not change.
The image has the same size as the original, although it may still be moved, rotated, or reflected.
Scale Factor Is Dimensionless
A scale factor compares two corresponding lengths:
\[ k=\frac{L'}{L}. \]Since both numerator and denominator have dimension \(L\),
\[ [k]=\frac{L}{L}=1. \]Scale factor is therefore dimensionless.
Corresponding Sides
To compare similar figures correctly, corresponding sides must be matched.
If
\[ \triangle ABC\sim\triangle DEF, \]then the order indicates the correspondence
\[ A\leftrightarrow D,\qquad B\leftrightarrow E,\qquad C\leftrightarrow F. \]Therefore,
\[ AB\leftrightarrow DE,\quad BC\leftrightarrow EF,\quad AC\leftrightarrow DF. \]Similarity Ratios
For similar figures, ratios of corresponding side lengths are equal.
If
\[ \triangle ABC\sim\triangle DEF, \]then
\[ \boxed{ \frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC} = k }. \]Example: Finding a Scale Factor
A side of length \(6\text{ cm}\) in the original corresponds to a side of length \(15\text{ cm}\) in the image.
Then
\[ k=\frac{15}{6}=\frac52. \] \[ \boxed{k=\frac52}. \]Example: Finding a New Length
A figure is enlarged by scale factor
\[ k=3. \]An original side has length \(8\text{ cm}\).
The corresponding image length is
\[ 3(8)=24. \] \[ \boxed{24\text{ cm}}. \]Example: Reduction
A drawing is reduced by scale factor
\[ k=\frac14. \]A \(20\text{ cm}\) segment becomes
\[ \frac14(20)=5. \] \[ \boxed{5\text{ cm}}. \]Finding an Unknown with a Proportion
Suppose corresponding sides satisfy
\[ \frac{x}{12}=\frac{15}{9}. \]Multiply:
\[ 9x=180. \]Therefore,
\[ \boxed{x=20}. \]Use Corresponding Sides in the Same Order
A similarity proportion must compare corresponding measurements in a consistent direction.
For example,
\[ \frac{\text{image}}{\text{original}} = \frac{\text{image}}{\text{original}}. \]Reversing only one ratio creates an invalid proportion.
Corresponding Angles Stay Equal
Uniform scaling changes length but does not change angle measure.
Therefore, if two figures are similar,
\[ \boxed{ \angle A\cong\angle D,\quad \angle B\cong\angle E,\quad \angle C\cong\angle F }. \]Shape is preserved because angular structure is preserved.
Dilations
A dilation is a transformation that scales distances from a fixed center by a common scale factor \(k\).
If \(O\) is the center and \(P\) maps to \(P'\), then
\[ \boxed{OP'=k\,OP}. \]Coordinate Dilations from the Origin
A dilation centered at the origin with scale factor \(k\) maps
\[ (x,y) \]to
\[ \boxed{(kx,ky)}. \]In three dimensions,
\[ (x,y,z)\mapsto(kx,ky,kz). \]Example: Coordinate Dilation
Dilate
\[ P=(3,-2) \]by scale factor \(4\) about the origin.
Multiply both coordinates by \(4\):
\[ P'=(12,-8). \] \[ \boxed{P'=(12,-8)}. \]Reduction in Coordinates
A point
\[ (10,6) \]dilated about the origin by
\[ k=\frac12 \]becomes
\[ \left(5,3\right). \]Dilations Preserve Lines through the Center
Under a dilation, a point and its image lie on the same ray from the center of dilation when \(k>0\).
A line passing through the center maps to itself as a set, although points on the line change position unless \(k=1\).
Lines Not through the Center
Under a dilation with positive scale factor, a line not passing through the center maps to a parallel line.
This preserves the directional structure of the figure while changing its size.
Scale Drawings
A scale drawing represents an object using proportional lengths.
For example, a plan may use
\[ 1\text{ cm}:2\text{ m}. \]This means every \(1\text{ cm}\) on the drawing represents \(2\text{ m}\) in the actual object.
Scale as a Ratio
A scale should compare lengths in compatible units when interpreted as a numerical ratio.
For
\[ 1\text{ cm}:2\text{ m}, \]convert \(2\text{ m}\) to \(200\text{ cm}\):
\[ 1:200. \]The drawing-to-actual scale factor is therefore
\[ \boxed{\frac1{200}}. \]Example: Reading a Scale Drawing
A map uses the scale
\[ 1\text{ cm}:5\text{ km}. \]Two locations are \(7.2\text{ cm}\) apart on the map.
The actual distance is
\[ 7.2(5)=36. \] \[ \boxed{36\text{ km}}. \]Models and Prototypes
Physical models also use scale factors.
A model built at
\[ 1:50 \]scale has linear dimensions equal to one-fiftieth of the corresponding dimensions of the original object.
Example: Model Length
A real structure is \(30\text{ m}\) long. A model is built at scale \(1:100\).
The model length is
\[ \frac{30\text{ m}}{100}=0.3\text{ m}. \] \[ \boxed{0.3\text{ m}=30\text{ cm}}. \]Indirect Measurement
Similarity can be used to measure lengths that are difficult to access directly.
If two geometric situations create similar triangles, corresponding side ratios can be used to solve for an unknown distance or height.
Example: Shadow Measurement
A \(1.8\text{ m}\) person casts a \(1.2\text{ m}\) shadow while a building casts an \(18\text{ m}\) shadow.
Assuming the sun's rays create similar triangles,
\[ \frac{1.8}{1.2} = \frac{h}{18}. \]Thus,
\[ 1.5=\frac{h}{18}. \] \[ h=27. \] \[ \boxed{h=27\text{ m}}. \]Similarity in Right Triangles
Similar triangles often arise inside right-triangle constructions.
Corresponding side ratios can reveal unknown lengths without requiring direct measurement.
This idea supports later work with trigonometry and geometric means.
Similarity in Coordinate Geometry
Similarity can be detected or constructed through coordinate relationships.
If every coordinate difference from a chosen center is multiplied by \(k\), corresponding distances are multiplied by \(k\).
Slopes of corresponding nonvertical sides remain unchanged under positive uniform scaling.
Why Uniform Scaling Preserves Shape
Uniform scaling multiplies every direction by the same factor.
Because all corresponding lengths change proportionally, relative side lengths remain unchanged:
\[ \frac{k a}{k b} = \frac{a}{b}. \]The common factor cancels.
Ratios within a Figure Are Preserved
Suppose two lengths in a figure satisfy
\[ \frac{a}{b}=\frac35. \]After scaling by \(k\), the corresponding ratio becomes
\[ \frac{ka}{kb} = \frac{a}{b} = \frac35. \]Uniform scaling changes magnitude but preserves internal proportions.
Perimeter under Scaling
If every side length is multiplied by \(k\), then perimeter is also multiplied by \(k\).
If
\[ P=a+b+c+\cdots, \]then after scaling,
\[ P'=ka+kb+kc+\cdots=kP. \] \[ \boxed{P'=kP}. \]Example: Perimeter Scaling
A polygon has perimeter \(42\text{ cm}\). It is enlarged by scale factor \(2.5\).
The new perimeter is
\[ 2.5(42)=105. \] \[ \boxed{105\text{ cm}}. \]Area Does Not Scale by the Same Factor
If every length is multiplied by \(k\), an area contains two scaled length dimensions.
For a rectangle,
\[ A'= (kl)(kw)=k^2lw. \]Therefore,
\[ \boxed{A'=k^2A}. \]The next lesson develops this scaling law systematically.
Volume Does Not Scale by the Same Factor
A three-dimensional volume contains three scaled length dimensions.
For a rectangular prism,
\[ V'=(kl)(kw)(kh)=k^3lwh. \]Therefore,
\[ \boxed{V'=k^3V}. \]Linear Similarity Ratio
The ratio
\[ \frac{\text{corresponding image length}} {\text{corresponding original length}} \]is the linear similarity ratio.
This ratio is the scale factor \(k\).
Area Similarity Ratio
If the linear similarity ratio is
\[ a:b, \]then the corresponding area ratio is
\[ \boxed{a^2:b^2}. \]This follows from two-dimensional scaling.
Volume Similarity Ratio
If the linear similarity ratio is
\[ a:b, \]then the corresponding volume ratio is
\[ \boxed{a^3:b^3}. \]This follows from three-dimensional scaling.
Example: From Length Ratio to Area Ratio
Two similar figures have corresponding side-length ratio
\[ 2:5. \]Their area ratio is
\[ 2^2:5^2. \] \[ \boxed{4:25}. \]Example: From Length Ratio to Volume Ratio
Two similar solids have corresponding length ratio
\[ 3:4. \]Their volume ratio is
\[ 3^3:4^3. \] \[ \boxed{27:64}. \]Recovering a Linear Ratio from an Area Ratio
If similar figures have area ratio
\[ 49:81, \]then the positive linear ratio is
\[ \sqrt{49}:\sqrt{81}. \] \[ \boxed{7:9}. \]Recovering a Linear Ratio from a Volume Ratio
If similar solids have volume ratio
\[ 8:125, \]then the corresponding positive linear ratio is
\[ \sqrt[3]{8}:\sqrt[3]{125}. \] \[ \boxed{2:5}. \]Similarity and Units
Corresponding lengths should be expressed in compatible units before a numerical scale factor is computed.
For example,
\[ \frac{2\text{ m}}{50\text{ cm}} \]should first be rewritten using one common unit:
\[ \frac{200\text{ cm}}{50\text{ cm}}=4. \] \[ \boxed{k=4}. \]Scale Factors and Unit Conversion Are Different
A unit conversion changes how the same physical length is expressed.
A scale factor changes the size of the geometric object or representation.
For example,
\[ 1\text{ m}=100\text{ cm} \]is a unit conversion, while enlarging a \(1\text{ m}\) segment to \(2\text{ m}\) uses scale factor \(2\).
Similarity Is Multiplicative
Scaling is naturally multiplicative.
If one transformation scales by \(a\) and a second scales by \(b\), the combined scale factor is
\[ \boxed{ab}. \]For example, scaling by \(2\) and then by \(3\) is equivalent to scaling by \(6\).
Inverse Scaling
A dilation by nonzero scale factor \(k\) can be reversed by a dilation with scale factor
\[ \boxed{\frac1k}. \]Their combined effect is
\[ k\left(\frac1k\right)=1. \]Similarity and Shape Invariants
Certain properties do not change under uniform scaling.
These include:
- angle measures;
- parallelism;
- perpendicularity;
- ratios of corresponding lengths within the same figure;
- overall shape.
Such preserved features are similarity invariants.
What Does Change under Scaling?
Quantities based on magnitude generally change.
Under scale factor \(k\):
\[ \text{length}\mapsto k(\text{length}), \] \[ \text{area}\mapsto k^2(\text{area}), \] \[ \text{volume}\mapsto k^3(\text{volume}). \]Dimensional structure determines the exponent.
Similarity in Higher Dimensions
The same principle extends to \(n\)-dimensional objects.
Uniform scaling by \(k\) multiplies every linear distance by \(k\), while \(n\)-dimensional measure scales by
\[ \boxed{k^n}. \]Common Error: Reversing the Scale Factor
If a \(4\text{ cm}\) side becomes \(10\text{ cm}\), the scale factor from original to image is
\[ \frac{10}{4}, \]not
\[ \frac{4}{10}. \]State the direction of comparison before forming the ratio.
Common Error: Comparing Noncorresponding Sides
A valid similarity ratio compares matching parts.
If the shortest side of one triangle corresponds to the shortest side of another, it should not be compared with the longest side merely because the numbers are convenient.
Establish correspondence first.
Common Error: Adding Instead of Multiplying
Scaling is multiplicative.
A scale factor of \(3\) changes a side of length \(5\) to
\[ 3(5)=15, \]not
\[ 5+3=8. \]Common Error: Using the Linear Ratio for Area
If corresponding lengths scale by \(3\), area does not merely triple.
Area scales by
\[ 3^2=9. \]Dimensional exponents control the scaling law.
Common Error: Using the Area Ratio for Volume
If the linear ratio is \(2:3\), the area ratio is
\[ 4:9, \]but the volume ratio is
\[ 8:27. \]The dimensional power must match the quantity being compared.
Common Error: Ignoring Units in a Scale Ratio
The ratio
\[ \frac{1\text{ cm}}{1\text{ m}} \]is not numerically \(1\).
Converting to common units gives
\[ \frac{1\text{ cm}}{100\text{ cm}} = \frac1{100}. \]Common Error: Assuming Equal Angles Alone Prove All Figures Similar
For triangles, angle-angle information is enough to establish similarity.
For general polygons, matching angles alone does not necessarily force corresponding side lengths to be proportional.
Similarity criteria depend on the class of figures being studied.
A Reliable Procedure for Finding a Scale Factor
- Identify the original figure and the image.
- Match a pair of corresponding lengths.
- Convert the lengths to compatible units if necessary.
- Compute \(\frac{\text{image}}{\text{original}}\).
- Check whether \(k>1\), \(0<k<1\), or \(k=1\).
- Verify the same ratio with another corresponding pair when possible.
A Reliable Procedure for Similarity Proportions
- Write the correspondence between the figures.
- Match corresponding sides carefully.
- Choose one consistent ratio direction.
- Set corresponding ratios equal.
- Solve the resulting proportion.
- Check that the answer agrees with the expected enlargement or reduction.
A Reliable Procedure for Scale Drawings
- Read the stated scale carefully.
- Identify whether the problem asks for drawing size or actual size.
- Convert units when a numerical ratio is needed.
- Multiply or divide by the appropriate scale factor.
- Attach the correct unit to the result.
- Check whether the answer is reasonable relative to the stated scale.
Example: Full Similarity Workflow
Two similar triangles have corresponding sides
\[ 6,\quad9,\quad12 \]and
\[ 10,\quad15,\quad x. \]From the first pair,
\[ k=\frac{10}{6}=\frac53. \]Check the second pair:
\[ \frac{15}{9}=\frac53. \]Therefore,
\[ x=\frac53(12)=20. \] \[ \boxed{x=20}. \]Example: Full Scale-Drawing Workflow
A blueprint uses
\[ 1\text{ cm}:0.5\text{ m}. \]A wall measures \(18\text{ cm}\) on the blueprint.
The actual wall length is
\[ 18(0.5)=9. \] \[ \boxed{9\text{ m}}. \]Practice 1: Find the Scale Factor
A side of length \(8\text{ cm}\) becomes \(20\text{ cm}\). Find the scale factor from original to image.
Practice 2: Find the Image Length
A segment of length \(14\text{ m}\) is enlarged by scale factor \(1.5\). Find the image length.
Practice 3: Reduction
A \(36\text{ cm}\) side is reduced by scale factor \(\frac13\). Find the new length.
Practice 4: Similar Figures
In two similar figures,
\[ \frac{x}{18}=\frac{10}{15}. \]Find \(x\).
Practice 5: Coordinate Dilation
Dilate
\[ (-4,6) \]by scale factor \(\frac12\) about the origin.
Practice 6: Map Scale
A map uses the scale \(1\text{ cm}:8\text{ km}\). Two cities are \(6.5\text{ cm}\) apart on the map. Find the actual distance.
Practice 7: Perimeter Scaling
A polygon has perimeter \(32\text{ cm}\) and is enlarged by scale factor \(3\). Find the new perimeter.
Practice 8: Area Ratio
Two similar figures have linear ratio
\[ 3:7. \]Find the ratio of their areas.
Practice 9: Volume Ratio
Two similar solids have linear ratio
\[ 2:5. \]Find the ratio of their volumes.
Practice 10: Recover the Linear Ratio
Two similar figures have area ratio
\[ 16:81. \]Find the positive linear ratio.
Practice Answers
Practice 1
\[ k=\frac{20}{8}=\frac52. \] \[ \boxed{k=\frac52}. \]Practice 2
\[ 1.5(14)=21. \] \[ \boxed{21\text{ m}}. \]Practice 3
\[ \frac13(36)=12. \] \[ \boxed{12\text{ cm}}. \]Practice 4
\[ \frac{x}{18}=\frac{10}{15}=\frac23. \] \[ x=18\left(\frac23\right)=12. \] \[ \boxed{x=12}. \]Practice 5
\[ \left(-4,6\right) \mapsto \left(-2,3\right). \] \[ \boxed{(-2,3)}. \]Practice 6
\[ 6.5(8)=52. \] \[ \boxed{52\text{ km}}. \]Practice 7
\[ 3(32)=96. \] \[ \boxed{96\text{ cm}}. \]Practice 8
\[ 3^2:7^2=9:49. \] \[ \boxed{9:49}. \]Practice 9
\[ 2^3:5^3=8:125. \] \[ \boxed{8:125}. \]Practice 10
\[ \sqrt{16}:\sqrt{81}=4:9. \] \[ \boxed{4:9}. \]What to Remember
- A scale factor compares corresponding image and original lengths.
- Similar figures have congruent corresponding angles and proportional corresponding lengths.
- \(k>1\) gives an enlargement, \(0<k<1\) gives a reduction, and \(k=1\) preserves size.
- Scale factor is dimensionless.
- Corresponding sides must be matched in a consistent order.
- A dilation centered at the origin maps \((x,y)\) to \((kx,ky)\).
- Scale drawings and models use similarity to represent larger or smaller objects.
- Perimeter scales by \(k\).
- Area scales by \(k^2\).
- Volume scales by \(k^3\).
The central principle is that uniform scaling multiplies every corresponding length by the same factor while preserving shape and internal proportions.
Where We Go Next
Scale factors describe how individual lengths change under similarity. The next lesson, Length, Area, and Volume Scaling Laws, develops the deeper dimensional consequences of that scaling.
It will explain why one-dimensional measures scale by \(k\), two-dimensional measures by \(k^2\), three-dimensional measures by \(k^3\), and \(n\)-dimensional measures by \(k^n\).