Similar Figures
Same shape, different size
Have you ever looked at a map and realized that the tiny shape representing your state is the exact same shape as your actual state - just much, much smaller? Or noticed how a photograph looks identical whether it is printed as a wallet size or blown up to poster size? That is the idea of similarity at work. Similar figures are shapes that look exactly alike - same angles, same proportions - but at different scales. Understanding similarity lets you solve problems involving scale models, blueprints, shadows, and countless real-world situations where you need to work with objects of different sizes that share the same shape.
Core Concepts
Ratio and Proportion: A Quick Review
Before we dive into similar figures, let us make sure we are comfortable with ratios and proportions, because they are the language of similarity.
A ratio compares two quantities. If a rectangle is 6 inches long and 4 inches wide, the ratio of length to width is $\frac{6}{4}$, which simplifies to $\frac{3}{2}$ or 3:2. This means the length is 1.5 times the width.
A proportion is an equation stating that two ratios are equal:
$$\frac{a}{b} = \frac{c}{d}$$
The key property of proportions is the cross-product property: if $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$. This gives us a powerful way to solve for unknown values.
For example, if $\frac{3}{4} = \frac{x}{20}$, then $3 \times 20 = 4 \times x$, so $60 = 4x$, and $x = 15$.
What Does “Similar” Mean in Geometry?
In everyday language, “similar” means “kind of alike.” In geometry, it has a very specific meaning:
Two figures are similar if:
- Their corresponding angles are congruent (equal in measure)
- Their corresponding sides are proportional (have the same ratio)
In other words, similar figures have the same shape but possibly different sizes. One figure is essentially a scaled-up or scaled-down version of the other.
We use the symbol $\sim$ to indicate similarity. If triangle ABC is similar to triangle DEF, we write:
$$\triangle ABC \sim \triangle DEF$$
The order of the letters matters! It tells you which vertices correspond to each other. In $\triangle ABC \sim \triangle DEF$:
- Angle A corresponds to angle D
- Angle B corresponds to angle E
- Angle C corresponds to angle F
- Side $\overline{AB}$ corresponds to side $\overline{DE}$
- Side $\overline{BC}$ corresponds to side $\overline{EF}$
- Side $\overline{AC}$ corresponds to side $\overline{DF}$
The Scale Factor
The scale factor is the ratio of any pair of corresponding sides between two similar figures. If $\triangle ABC \sim \triangle DEF$, then:
$$k = \frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC}$$
All three ratios are equal - that is what it means for the sides to be proportional.
If the scale factor $k = 2$, it means every side of the second figure is twice as long as the corresponding side of the first figure. If $k = \frac{1}{3}$, the second figure is one-third the size of the first.
Important: The scale factor depends on which figure you put in the numerator. The scale factor from $\triangle ABC$ to $\triangle DEF$ might be 2, while the scale factor from $\triangle DEF$ to $\triangle ABC$ is $\frac{1}{2}$. These are reciprocals of each other.
Similar Polygons
Everything we have said about similar triangles applies to any similar polygons. Two polygons are similar if:
- All pairs of corresponding angles are congruent
- All pairs of corresponding sides are proportional
For quadrilaterals, you need to check four pairs of angles and four ratios of sides. For pentagons, five of each, and so on. But the core idea remains the same: same shape, different size.
Proving Triangles Similar: The AA Similarity Postulate
With triangles, we get a wonderful shortcut. We do not actually need to check all three pairs of angles AND all three pairs of sides. The AA (Angle-Angle) Similarity Postulate states:
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Why does this work? Remember that the angles in any triangle sum to $180°$. If two angles match, the third angle must also match (since all three must add to $180°$). And when all three angles match, the triangles have the same shape - they must be similar.
This is incredibly useful. To prove two triangles are similar, you only need to show that two pairs of corresponding angles are equal.
Proving Triangles Similar: The SSS Similarity Theorem
The SSS (Side-Side-Side) Similarity Theorem provides another way to prove similarity:
If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar.
In other words, if $\frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC}$, then $\triangle ABC \sim \triangle DEF$.
You do not need to know any angles - if all three ratios of corresponding sides are equal, the triangles are similar.
Proving Triangles Similar: The SAS Similarity Theorem
The SAS (Side-Angle-Side) Similarity Theorem is a third option:
If two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the triangles are similar.
The “included angle” is the angle formed by the two sides you are comparing. So if $\frac{AB}{DE} = \frac{AC}{DF}$ and $\angle A \cong \angle D$, then $\triangle ABC \sim \triangle DEF$.
Finding Missing Sides in Similar Figures
Once you know two figures are similar, you can use proportions to find missing side lengths. Set up a proportion using corresponding sides, making sure to match the figures consistently.
For example, if $\triangle ABC \sim \triangle DEF$, and you know $AB = 6$, $DE = 9$, and $BC = 8$, you can find $EF$:
$$\frac{AB}{DE} = \frac{BC}{EF}$$
$$\frac{6}{9} = \frac{8}{EF}$$
Cross-multiply: $6 \times EF = 9 \times 8$
$6 \times EF = 72$
$EF = 12$
Perimeters of Similar Figures
If two figures are similar with scale factor $k$, their perimeters also have a ratio of $k$.
Think about it: if every side of the second figure is $k$ times the corresponding side of the first figure, then when you add up all the sides (the perimeter), the second perimeter is also $k$ times the first perimeter.
$$\frac{\text{Perimeter of larger figure}}{\text{Perimeter of smaller figure}} = k$$
Areas of Similar Figures
Here is where things get interesting. If two similar figures have a scale factor of $k$, the ratio of their areas is $k^2$.
Why? Area is measured in square units, so it scales by the square of the linear scale factor. If you double the sides of a square, you quadruple its area. If you triple the sides, you multiply the area by nine.
$$\frac{\text{Area of larger figure}}{\text{Area of smaller figure}} = k^2$$
This is a crucial relationship. If a photo is enlarged by a scale factor of 3, it covers $3^2 = 9$ times as much wall space.
Notation and Terminology
| Term | Meaning | Example |
|---|---|---|
| Similar | Same shape, proportional sides, congruent angles | $\triangle ABC \sim \triangle DEF$ |
| Scale factor | Ratio of corresponding sides | $k = 2$ means twice as large |
| Proportion | An equation stating two ratios are equal | $\frac{a}{b} = \frac{c}{d}$ |
| Corresponding | Matching parts in similar figures | $\angle A$ corresponds to $\angle D$ |
| AA Similarity | Two pairs of congruent angles prove similarity | Angles A=D and B=E mean similar |
| SSS Similarity | Three proportional side pairs prove similarity | All ratios equal means similar |
| SAS Similarity | Two proportional sides with included angle | Matching ratio and included angle |
| Congruent | Equal in measure | $\angle A \cong \angle B$ means equal angles |
Examples
Rectangle ABCD has length 12 cm and width 8 cm. Rectangle EFGH has length 9 cm and width 6 cm. Are these rectangles similar?
Solution:
For rectangles to be similar, their corresponding sides must be proportional. All angles in rectangles are $90°$, so angles automatically match.
Check if the ratios of corresponding sides are equal:
$$\frac{\text{length of ABCD}}{\text{length of EFGH}} = \frac{12}{9} = \frac{4}{3}$$
$$\frac{\text{width of ABCD}}{\text{width of EFGH}} = \frac{8}{6} = \frac{4}{3}$$
Both ratios equal $\frac{4}{3}$, so the sides are proportional.
Yes, the rectangles are similar with scale factor $\frac{4}{3}$ (or $\frac{3}{4}$ going from ABCD to EFGH).
Note: Not all rectangles are similar! A 12 by 8 rectangle is not similar to a 12 by 4 rectangle because $\frac{12}{12} = 1$ but $\frac{8}{4} = 2$. The ratios do not match.
$\triangle ABC \sim \triangle XYZ$. Given $AB = 5$, $BC = 7$, $AC = 8$, and $XY = 15$, find $YZ$ and $XZ$.
Solution:
First, find the scale factor using the known corresponding sides $AB$ and $XY$:
$$k = \frac{XY}{AB} = \frac{15}{5} = 3$$
The second triangle is 3 times as large as the first.
Now find the missing sides:
$$YZ = BC \times k = 7 \times 3 = 21$$
$$XZ = AC \times k = 8 \times 3 = 24$$
The missing sides are $\boxed{YZ = 21}$ and $\boxed{XZ = 24}$.
Check: Verify that all ratios are equal:
- $\frac{XY}{AB} = \frac{15}{5} = 3$
- $\frac{YZ}{BC} = \frac{21}{7} = 3$
- $\frac{XZ}{AC} = \frac{24}{8} = 3$
All ratios equal 3. Confirmed!
Two similar pentagons have perimeters of 35 inches and 56 inches. What is the scale factor from the smaller pentagon to the larger pentagon? If one side of the smaller pentagon is 9 inches, what is the corresponding side of the larger pentagon?
Solution:
Since the pentagons are similar, the ratio of their perimeters equals the scale factor.
$$k = \frac{\text{Perimeter of larger}}{\text{Perimeter of smaller}} = \frac{56}{35} = \frac{8}{5} = 1.6$$
The scale factor is $\boxed{k = \frac{8}{5}}$ (or 1.6).
To find the corresponding side of the larger pentagon:
$$\text{Larger side} = \text{Smaller side} \times k = 9 \times \frac{8}{5} = \frac{72}{5} = 14.4 \text{ inches}$$
The corresponding side of the larger pentagon is $\boxed{14.4 \text{ inches}}$.
In the figure, $\overline{DE}$ is parallel to $\overline{BC}$. Prove that $\triangle ADE \sim \triangle ABC$, and find $DE$ if $AD = 4$, $AB = 10$, and $BC = 15$.
Solution:
Proving Similarity:
Since $\overline{DE} \parallel \overline{BC}$, we know that:
- $\angle ADE \cong \angle ABC$ (corresponding angles formed by parallel lines and transversal $\overline{AB}$)
- $\angle AED \cong \angle ACB$ (corresponding angles formed by parallel lines and transversal $\overline{AC}$)
- $\angle A \cong \angle A$ (the same angle, shared by both triangles)
We have two pairs of congruent angles (actually three, but we only need two). By the AA Similarity Postulate, $\triangle ADE \sim \triangle ABC$.
Finding DE:
Since the triangles are similar, corresponding sides are proportional:
$$\frac{AD}{AB} = \frac{DE}{BC}$$
$$\frac{4}{10} = \frac{DE}{15}$$
Cross-multiply:
$$10 \times DE = 4 \times 15$$
$$10 \times DE = 60$$
$$DE = 6$$
Therefore, $\boxed{DE = 6}$.
Interpretation: Since $\frac{AD}{AB} = \frac{4}{10} = \frac{2}{5}$, point D is $\frac{2}{5}$ of the way from A to B. The segment DE is also $\frac{2}{5}$ as long as BC.
Two similar triangles have a scale factor of $\frac{3}{5}$ (smaller to larger). The area of the smaller triangle is 27 square centimeters. What is the area of the larger triangle?
Solution:
For similar figures, the ratio of areas equals the square of the scale factor.
The scale factor from smaller to larger is $\frac{3}{5}$, so:
$$\frac{\text{Area of smaller}}{\text{Area of larger}} = \left(\frac{3}{5}\right)^2 = \frac{9}{25}$$
Now we can set up a proportion:
$$\frac{27}{\text{Area of larger}} = \frac{9}{25}$$
Cross-multiply:
$$27 \times 25 = 9 \times \text{Area of larger}$$
$$675 = 9 \times \text{Area of larger}$$
$$\text{Area of larger} = 75 \text{ square centimeters}$$
The area of the larger triangle is $\boxed{75 \text{ cm}^2}$.
Alternative approach: Since the scale factor (smaller to larger) is $\frac{3}{5}$, the larger triangle’s sides are $\frac{5}{3}$ times the smaller triangle’s sides. The area ratio is $\left(\frac{5}{3}\right)^2 = \frac{25}{9}$.
Area of larger = $27 \times \frac{25}{9} = \frac{27 \times 25}{9} = 3 \times 25 = 75$ square cm.
Same answer, different route!
A 6-foot-tall person casts a 4-foot shadow at the same time that a tree casts a 22-foot shadow. How tall is the tree?
Solution:
The sun’s rays hit both the person and the tree at the same angle (since the shadows are measured at the same time). This creates two similar right triangles:
- One formed by the person, their shadow, and the sun’s ray
- One formed by the tree, its shadow, and the sun’s ray
Both triangles have:
- A right angle (where the object meets the ground)
- The same angle of elevation for the sun’s rays
By AA Similarity, the triangles are similar.
Let $h$ = height of the tree. Setting up a proportion with corresponding sides:
$$\frac{\text{Person’s height}}{\text{Person’s shadow}} = \frac{\text{Tree’s height}}{\text{Tree’s shadow}}$$
$$\frac{6}{4} = \frac{h}{22}$$
Cross-multiply:
$$4h = 6 \times 22$$
$$4h = 132$$
$$h = 33$$
The tree is $\boxed{33 \text{ feet}}$ tall.
This is the power of similar figures: you can measure the height of something you cannot reach by using shadows and proportions!
Key Properties and Rules
Definition of Similarity
Two polygons are similar if and only if:
- All pairs of corresponding angles are congruent
- All pairs of corresponding sides are proportional
$$\triangle ABC \sim \triangle DEF \iff \begin{cases} \angle A \cong \angle D, \ \angle B \cong \angle E, \ \angle C \cong \angle F \ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \end{cases}$$
Triangle Similarity Shortcuts
AA Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
SSS Similarity Theorem: If three pairs of corresponding sides are proportional, the triangles are similar.
SAS Similarity Theorem: If two pairs of corresponding sides are proportional and the included angles are congruent, the triangles are similar.
Scale Factor Relationships
If two similar figures have scale factor $k$:
Ratio of corresponding sides: $k$
Ratio of perimeters: $k$
Ratio of areas: $k^2$
Ratio of volumes (for 3D similar figures): $k^3$
Cross-Product Property of Proportions
If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$.
This is the fundamental tool for solving proportions.
Properties of Proportions
If $\frac{a}{b} = \frac{c}{d}$, then:
- $\frac{b}{a} = \frac{d}{c}$ (inverting both sides)
- $\frac{a}{c} = \frac{b}{d}$ (alternating)
- $\frac{a + b}{b} = \frac{c + d}{d}$ (adding the denominator to the numerator)
Real-World Applications
Scale Models and Maps
Maps are the ultimate example of similarity. A map of the United States is similar to the actual shape of the country - same outline, same angles, just scaled way down. The map legend tells you the scale factor: “1 inch = 50 miles” means every inch on the map represents 50 miles in reality.
Architects and engineers use scale models to visualize buildings before construction. A model at 1:100 scale means every centimeter on the model represents 100 centimeters (1 meter) in the real building. The angles and proportions are preserved exactly.
Photograph Enlargement
When you enlarge a photograph from 4x6 inches to 8x12 inches, you are creating a similar figure. The scale factor is 2. The image looks identical - same proportions, same angles - just bigger.
But watch out: a 4x6 photo is not similar to a 5x7 photo. The ratio $\frac{4}{6} = \frac{2}{3}$ does not equal $\frac{5}{7}$. If you try to print a 4x6 image as 5x7, part of the image will be cropped or the proportions will be distorted.
Shadow Problems and Indirect Measurement
We saw this in Example 6. Similar triangles let us measure heights we cannot reach directly. Surveyors have used this technique for thousands of years to measure mountains, buildings, and distances across rivers.
The key insight: parallel rays of light (from the sun or a distant light source) create similar triangles when they hit objects of different heights.
Architecture and Construction
Blueprints are similar figures of the actual building. Every length on the blueprint corresponds proportionally to a length in the real structure. This allows architects to design on paper and builders to construct at full scale while maintaining all the correct proportions.
Art and Design
Artists use the concept of similar figures when scaling drawings. If you want to enlarge a small sketch onto a large canvas while keeping everything in proportion, you divide both the original and the canvas into grids with the same number of squares. Each small square is similar to each large square, making it easy to transfer the image proportionally.
Forensics and Crime Scene Investigation
Crime scene investigators use similar triangles to calculate trajectories. If a bullet hole is at a certain height and angle, similar triangles help determine where the shot was fired from.
Photography and Cameras
The lens in a camera creates an image that is similar to the actual scene. The focal length and distance to the object determine the scale factor. Understanding this similarity is essential for calculating image sizes, depth of field, and more.
Self-Test Problems
Problem 1: $\triangle PQR \sim \triangle XYZ$. If $PQ = 8$, $QR = 12$, $PR = 10$, and $XY = 20$, find $YZ$ and $XZ$.
Show Answer
Find the scale factor: $$k = \frac{XY}{PQ} = \frac{20}{8} = \frac{5}{2} = 2.5$$
Find the missing sides: $$YZ = QR \times k = 12 \times 2.5 = 30$$ $$XZ = PR \times k = 10 \times 2.5 = 25$$
The missing sides are $YZ = 30$ and $XZ = 25$.
Problem 2: Are the triangles with sides 6, 8, 10 and sides 9, 12, 15 similar? If so, what is the scale factor?
Show Answer
Check if all three ratios of corresponding sides are equal:
$$\frac{9}{6} = \frac{3}{2}$$ $$\frac{12}{8} = \frac{3}{2}$$ $$\frac{15}{10} = \frac{3}{2}$$
All three ratios equal $\frac{3}{2}$.
Yes, the triangles are similar (by SSS Similarity). The scale factor is $\frac{3}{2}$ or 1.5.
(Note: Both triangles are also right triangles - they are 3-4-5 and 9-12-15 right triangles!)
Problem 3: In $\triangle ABC$, $\angle A = 50°$ and $\angle B = 60°$. In $\triangle DEF$, $\angle D = 50°$ and $\angle F = 70°$. Are these triangles similar? Explain.
Show Answer
Find all angles:
In $\triangle ABC$: $\angle C = 180° - 50° - 60° = 70°$
In $\triangle DEF$: $\angle E = 180° - 50° - 70° = 60°$
Now compare:
- $\angle A = 50° = \angle D$
- $\angle B = 60° = \angle E$
- $\angle C = 70° = \angle F$
All three pairs of corresponding angles are congruent.
Yes, the triangles are similar by AA Similarity (we only needed two pairs, but all three match).
$\triangle ABC \sim \triangle DEF$
Problem 4: Two similar rectangles have perimeters of 24 cm and 36 cm. If the smaller rectangle has an area of 32 square cm, what is the area of the larger rectangle?
Show Answer
Find the scale factor from perimeters: $$k = \frac{36}{24} = \frac{3}{2} = 1.5$$
The ratio of areas is $k^2$: $$k^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$$
Find the larger area: $$\frac{\text{Area of larger}}{\text{Area of smaller}} = \frac{9}{4}$$ $$\frac{\text{Area of larger}}{32} = \frac{9}{4}$$ $$\text{Area of larger} = 32 \times \frac{9}{4} = 8 \times 9 = 72 \text{ square cm}$$
The area of the larger rectangle is 72 square cm.
Problem 5: A flagpole casts a shadow of 18 feet at the same time a 5-foot-tall student casts a shadow of 3 feet. How tall is the flagpole?
Show Answer
The flagpole and student form similar triangles with their shadows (same sun angle).
Set up a proportion: $$\frac{\text{Flagpole height}}{\text{Flagpole shadow}} = \frac{\text{Student height}}{\text{Student shadow}}$$ $$\frac{h}{18} = \frac{5}{3}$$
Cross-multiply: $$3h = 5 \times 18$$ $$3h = 90$$ $$h = 30$$
The flagpole is 30 feet tall.
Problem 6: $\triangle ABC \sim \triangle DEF$ with scale factor 4:7 (ABC to DEF). If the perimeter of $\triangle ABC$ is 28 cm, what is the perimeter of $\triangle DEF$?
Show Answer
For similar figures, the ratio of perimeters equals the scale factor.
$$\frac{\text{Perimeter of ABC}}{\text{Perimeter of DEF}} = \frac{4}{7}$$
$$\frac{28}{\text{Perimeter of DEF}} = \frac{4}{7}$$
Cross-multiply: $$4 \times \text{Perimeter of DEF} = 28 \times 7$$ $$4 \times \text{Perimeter of DEF} = 196$$ $$\text{Perimeter of DEF} = 49 \text{ cm}$$
The perimeter of $\triangle DEF$ is 49 cm.
Problem 7: The ratio of the areas of two similar triangles is 16:49. What is the ratio of their corresponding sides?
Show Answer
For similar figures, the ratio of areas equals the square of the ratio of sides.
If the area ratio is $\frac{16}{49}$, then: $$k^2 = \frac{16}{49}$$ $$k = \sqrt{\frac{16}{49}} = \frac{\sqrt{16}}{\sqrt{49}} = \frac{4}{7}$$
The ratio of corresponding sides is 4:7.
Summary
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Similar figures have the same shape but different sizes. This means all corresponding angles are congruent and all corresponding sides are proportional.
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The scale factor $k$ is the ratio of any pair of corresponding sides. It tells you how many times larger (or smaller) one figure is compared to the other.
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To prove triangles similar, you can use:
- AA Similarity: Two pairs of congruent angles
- SSS Similarity: Three pairs of proportional sides
- SAS Similarity: Two pairs of proportional sides with congruent included angles
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To find missing sides in similar figures, set up a proportion using corresponding sides and cross-multiply to solve.
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The perimeter ratio of similar figures equals the scale factor $k$.
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The area ratio of similar figures equals $k^2$ (the square of the scale factor).
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Similar figures appear everywhere in real life: maps, photographs, blueprints, scale models, and shadow problems. Any time you need to work with the same shape at different sizes, similarity is your tool.
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The order of letters in a similarity statement matters! $\triangle ABC \sim \triangle DEF$ tells you exactly which vertices correspond to each other.