Similar Figures and Scale Factor

🔺 Similar Figures & Scale Factor

Part 1 of 5 — What Similarity Means

Topics in This Part

🔑 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

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

1. Corresponding angles are equal (the shape matches).
2. 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:

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).

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:

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

Example

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.

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

Find the Scale Factor 🧮

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

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)

Enlargement or Reduction? 🔽

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

🔺 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 $AB = 8$ matching $DE = 12$, and $BC = 6$. Find $EF$.

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

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

1. Cross-multiply: $9 \cdot x = 3 \cdot 15$
2. Simplify: $9x = 45$
3. 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 $x$.

🔺 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.

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

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

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 $> 1$).

🔺 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:

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

Example

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 "$1:24$".

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.

🔺 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

⚠️ 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.