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Similar Figures and Scale Factor

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Similar Figures and Scale Factor - Complete Interactive Lesson

Part 1: What Similarity Means

🔺 Similar Figures & Scale Factor

Part 1 of 5 — What Similarity Means

Topics in This Part

Section
Congruent vs. Similar
Corresponding Parts
The Scale Factor

🔑 Key Concept: Two figures are similar when they have the same shape but not necessarily the same size. One is a scaled copy of the other — like a photo and its enlargement.

Congruent vs. Similar

	Same shape?	Same size?	Symbol
Congruent	yes	yes	$\cong$
Similar	yes	no (any size)	$\sim$

Two polygons are similar ( $\sim$ ) when both of these are true:

Corresponding angles are equal (the shape matches).
Corresponding sides are proportional (all scaled by the same number).

$\triangle ABC \sim \triangle DEF$

This statement is read "triangle $ABC$ is similar to triangle $DEF$ ." The order of the letters matters — it tells you which parts correspond:

$A \leftrightarrow D, \quad B \leftrightarrow E, \quad C \leftrightarrow F$

⚠️ Congruent figures are also similar (a scale factor of $1$ ). But similar figures are usually not congruent.

Corresponding Parts

When you write $\triangle ABC \sim \triangle DEF$ , line up the letters to match parts:

Corresponding angles	Corresponding sides
$\angle A = \angle D$	$A B \leftrightarrow$

Concept Check 🎯

The Scale Factor

The scale factor ( $k$ ) is the single number that every side is multiplied by to get from one figure to the other:

$k = \frac{\text{new (image) side}}{\text{original side}} = \frac{\text{a side of figure 2}}{\text{corresponding side of figure 1}}$

Find the Scale Factor 🧮

For each pair of corresponding sides, find $k$ going from figure 1 to figure 2 (so $k = \dfrac{\text{side}_2}{\text{side}_1}$ ).

Enlargement or Reduction? 🔽

For each scale factor (figure 1 → figure 2), choose what happens to the figure.

Part 2: Finding Missing Sides

🔺 Similar Figures & Scale Factor

Part 2 of 5 — Finding Missing Sides

🔑 The Idea: Because corresponding sides are proportional, a single missing length can be found by setting up a proportion or by multiplying by the scale factor.

Two Ways to Find a Missing Side

Method 1 — Scale factor

Find $k$ from a known pair of corresponding sides, then multiply.

Example: $\triangle ABC \sim \triangle DEF$ with $A$ matching , and . Find .

Part 3: Perimeter, Area & Volume Ratios

🔺 Similar Figures & Scale Factor

Part 3 of 5 — Perimeter, Area & Volume Ratios

🔑 The Big Pattern: If the scale factor for lengths is $k$ , then perimeter scales by $k$ , area scales by $k^2$ , and volume scales by $k^3$ . Dimensions become exponents.

Part 4: Real-World Applications

🔺 Similar Figures & Scale Factor

Part 4 of 5 — Real-World Applications

🔑 Big Payoff: Similarity lets you measure things you can't reach — the height of a tree, the distance on a map, the real size of a model — all with a single proportion.

Indirect Measurement (Shadows)

At the same time of day, the sun makes similar triangles out of every object and its shadow. So:

$\frac{\text{object 1 height}}{\text{object 1 shadow}} = \frac{\text{object 2 height}}{\text{object 2 shadow}}$

Part 5: Mixed Practice & Mastery Check

🔺 Similar Figures & Scale Factor

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) recognize and name similar figures, (2) find a scale factor, (3) solve for missing sides, (4) scale perimeter/area/volume by $k$ , $k^2$ , $k^3$ , and (5) apply similarity to the real world. Let's put it together.

Quick Reference

A B \leftrightarrow D E

The proportional-side rule says all the side ratios are equal:

$\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$

💡 To find a side that corresponds to another, read the same position in each name. In $ABCD \sim WXYZ$ , side $BC$ corresponds to side $XY$ (2nd and 3rd letters).

corresponding side of figure 1a side of figure 2

A small triangle has a side of length $4$ . The corresponding side in the larger, similar triangle is $12$ .

$k = \frac{12}{4} = 3$

Every side of the big triangle is 3 times the matching side of the small one.

Scale factor $k$	Meaning
$k > 1$	enlargement (image is bigger)
$k = 1$	congruent (same size)
$0 < k < 1$	reduction (image is smaller)

🔑 Key Idea: $k$ is just the ratio of corresponding sides. Once you know $k$ , you can find any missing side by multiplying (or dividing).

1) side $_1 = 5$ , side $_2 = 20 \;\Rightarrow\; k = \,?$ 2) side $_1 = 6$ , side $_2 = 9 \;\Rightarrow\; k = \,?$ (decimal is fine) 3) side $_1 = 10$ , side $_2 = 4 \;\Rightarrow\; k = \,?$ (decimal is fine)

$k = \frac{DE}{AB} = \frac{12}{8} = 1.5 \quad\Rightarrow\quad EF = BC \cdot k = 6 \cdot 1.5 = 9$

Method 2 — Proportion

Set up matching ratios and cross-multiply.

$\frac{AB}{DE} = \frac{BC}{EF} \;\Rightarrow\; \frac{8}{12} = \frac{6}{EF}$

Cross-multiply: $8 \cdot EF = 12 \cdot 6 = 72$ , so $EF = \dfrac{72}{8} = 9$ . ✓

💡 Both methods always agree. Use a proportion when the unknown is inside a ratio; use the scale factor when you'll find several sides at once.

Cross-Multiplication, Step by Step

To solve $\dfrac{3}{x} = \dfrac{9}{15}$ :

Cross-multiply: $9 \cdot x = 3 \cdot 15$
Simplify: $9x = 45$
Divide: $x = \dfrac{45}{9} = 5$

⚠️ Set up ratios consistently. Keep figure 1 on top and figure 2 on the bottom in both fractions (or vice versa). Mixing the order is the #1 mistake.

Concept Check 🎯

Build the Proportion 🔽

You're solving $\dfrac{3}{x} = \dfrac{9}{15}$ for $x$ . Choose what happens at each stage.

Solve for the Missing Side 🧮

Each pair of figures is similar. Find the missing length.

1) $\triangle ABC \sim \triangle DEF$ . $AB = 8 \leftrightarrow DE = 12$ , and $BC = 6$ . Find $EF$ . 2) Similar rectangles: small is $4 \times 10$ , large width $= 6$ (matches the $4$ ). Find the large length. 3) $\dfrac{5}{x} = \dfrac{15}{21}$ . Find $x$ .

The $k$ , $k^2$ , $k^3$ Rule

Quantity	Dimension	Ratio (figure 2 : figure 1)
Side / Perimeter	length (1-D)	$k$
Area / Surface area	area (2-D)	$k^2$
Volume	volume (3-D)	$k^3$

Why area uses $k^2$

A square with side $s$ has area $s^2$ . Scale the side by $k$ :

$\text{new area} = (ks)^2 = k^2 s^2 = k^2 \cdot (\text{old area})$

The same logic gives volume $= (ks)^3 = k^3 s^3$ .

Example

Two similar triangles have scale factor $k = 3$ .

Perimeter ratio $= 3$
Area ratio $= 3^2 = 9$ (the big one has 9× the area)

⚠️ A common trap: doubling the side does not double the area — it makes the area $2^2 = 4$ times larger.

Working Backwards

You can also go from an area or volume ratio back to the length scale factor:

$k = \sqrt{\text{area ratio}} \qquad k = \sqrt[3]{\text{volume ratio}}$

Example

Two similar pentagons have areas $25\text{ cm}^2$ and $100\text{ cm}^2$ .

$\text{area ratio} = \frac{100}{25} = 4 \quad\Rightarrow\quad k = \sqrt{4} = 2$

So the larger pentagon's sides are only $2\times$ as long, even though its area is $4\times$ as big.

💡 Reduction check: for a half-size copy ( $k = \frac12$ ), the area ratio is $\left(\frac12\right)^2 = \frac14$ and the volume ratio is .

Concept Check 🎯

Match the Ratio 🔽

For each scaling situation, choose the correct multiplier.

Scale the Measurements 🧮

1) A figure has perimeter $14$ . A similar figure has $k = 2.5$ . Find its perimeter. 2) A figure has area $5\text{ cm}^2$ . A similar figure has $k = 2$ . Find its area (cm²). 3) A solid has volume $8\text{ cm}^3$ . A similar solid has $k = 3$ . Find its volume (cm³). 4) Two similar figures have areas $100$ and $25$ . Find the scale factor $k$ (larger → smaller is not asked — give large:small as a number $> 1$ ).

object 2 shadowobject 2 height

A $6$ -ft person casts a $4$ -ft shadow. At the same moment, a tree casts a $24$ -ft shadow. How tall is the tree?

$\frac{6}{4} = \frac{h}{24} \;\Rightarrow\; 4h = 6 \cdot 24 = 144 \;\Rightarrow\; h = \frac{144}{4} = 36 \text{ ft}$

💡 Keep the same kind of measurement on top in both ratios (height over shadow = height over shadow). The triangles are similar because the sun's angle is identical.

Scale Drawings, Maps & Models

A scale is a ratio that compares a drawing/model to the real thing, e.g. " $1\text{ in} = 50\text{ mi}$ " or " $1:24$ ".

Scale	Meaning
$1\text{ in} = 50\text{ mi}$ (map)	each inch on the map is $50$ real miles
$1:24$ (model car)	the real car is $24\times$ the model
$1\text{ in} = 4\text{ ft}$ (blueprint)	each inch on paper is $4$ real feet

Example — Map

Two cities are $3.5$ in apart on a map scaled $1\text{ in} = 50\text{ mi}$ .

$\text{real distance} = 3.5 \times 50 = 175 \text{ mi}$

Example — Model

A model car is built at $1:24$ . The model is $8$ in long. The real car is:

$8 \times 24 = 192 \text{ in} = \frac{192}{12} = 16 \text{ ft}$

⚠️ Watch your units. Convert at the end (here $192$ inches $\to 16$ feet), not in the middle of the proportion.

Concept Check 🎯

Set Up the Map Problem 🔽

A map uses the scale $1\text{ in} = 50\text{ mi}$ . Two cities are $3.5$ in apart.

Apply It 🧮

1) A $6$ -ft person casts a $4$ -ft shadow; a tree casts a $24$ -ft shadow. Find the tree's height (ft). 2) Map scale $1\text{ in} = 50\text{ mi}$ . Two towns are $3.5$ in apart. Find the real distance (mi). 3) A model is built at $1:24$ and is $8$ in long. Find the real length in feet.

⚠️ Remember: the order of letters in $\triangle ABC \sim \triangle DEF$ tells you which parts correspond, and area scales faster than length (squared), volume faster still (cubed).

Mixed Practice 🎯

One More Set 🔽

Exit Quiz ✅

Answer all three to finish the lesson.

\left(\frac12\right)^3 = \frac18

Test for similarity	equal angles and proportional sides
Scale factor $k$	$\dfrac{\text{new side}}{\text{original side}}$
Missing side	proportion + cross-multiply
Perimeter ratio	$k$
Area ratio	$k^2$
Volume ratio	$k^3$
Back out $k$	$\sqrt{\text{area ratio}}$ or $\sqrt[3]{\text{volume ratio}}$

Similar Figures and Scale Factor

Similar Figures and Scale Factor - Complete Interactive Lesson

Part 1: What Similarity Means

🔺 Similar Figures & Scale Factor

Topics in This Part

Congruent vs. Similar

Corresponding Parts

The Scale Factor

Part 2: Finding Missing Sides

🔺 Similar Figures & Scale Factor

Two Ways to Find a Missing Side

Method 1 — Scale factor

Part 3: Perimeter, Area & Volume Ratios

🔺 Similar Figures & Scale Factor

Part 4: Real-World Applications

🔺 Similar Figures & Scale Factor

Indirect Measurement (Shadows)

Part 5: Mixed Practice & Mastery Check

🔺 Similar Figures & Scale Factor

Quick Reference

Example

Method 2 — Proportion

Cross-Multiplication, Step by Step

The kkk, k2k^2k2, k3k^3k3 Rule

Why area uses k2k^2k2

Example

Working Backwards

Example

Example

Scale Drawings, Maps & Models

Example — Map

Example — Model

The $k$ , $k^2$ , $k^3$ Rule

Why area uses $k^2$