Using Scale Factor To Find Length Of Line Segments

🔑 Key Concepts

• Similar figures have congruent corresponding angles and proportional corresponding side lengths.
• The larger figure and the smaller figure must use one consistent scale factor.
• The side \(\overline{AB}\) corresponds to \(\overline{GH}\).
• Because \(AB=10\) cm and \(GH=6\) cm, the scale factor from the larger figure to the smaller figure is \(\frac{6}{10}=\frac{3}{5}\).
• The side \(\overline{DE}\) corresponds to \(\overline{JK}\).
• The side \(\overline{FE}\) corresponds to \(\overline{LK}\).
• Multiply each corresponding side of the larger figure by \(\frac{3}{5}\) to find the matching side in the smaller figure.

✏️ Worked Example

Problem

The two polygons shown are similar figures. In the larger figure, \(\overline{AB}=10\) centimeters, \(\overline{DE}=3\) centimeters, and \(\overline{FE}=7\) centimeters. In the smaller figure, the corresponding side \(\overline{GH}=6\) centimeters. Find the lengths of \(\overline{JK}\) and \(\overline{LK}\).

Because the figures are similar, all corresponding side lengths are related by the same scale factor.

Step 1: Find the Scale Factor

The corresponding vertical sides are \(\overline{AB}\) and \(\overline{GH}\).

\(AB=10\text{ cm}\)

\(GH=6\text{ cm}\)

The scale factor from the larger figure to the smaller figure is:

\(\displaystyle k=\frac{\text{smaller}}{\text{larger}}\)

\(\displaystyle k=\frac{6}{10}\)

\(\displaystyle k=\frac{3}{5}=0.6\)

\(\boxed{k=\frac{3}{5}}\)

Step 2: Find \(\overline{JK}\)

The bottom horizontal side \(\overline{DE}\) corresponds to \(\overline{JK}\).

\(DE=3\text{ cm}\)

Multiply by the scale factor:

\(\displaystyle JK=\frac{3}{5}(DE)\)

\(\displaystyle JK=\frac{3}{5}(3)\)

\(\displaystyle JK=\frac{9}{5}\)

\(JK=1.8\)

\(\boxed{JK=1.8\text{ cm}}\)

Step 3: Find \(\overline{LK}\)

The right vertical side \(\overline{FE}\) corresponds to \(\overline{LK}\).

\(FE=7\text{ cm}\)

Multiply by the same scale factor:

\(\displaystyle LK=\frac{3}{5}(FE)\)

\(\displaystyle LK=\frac{3}{5}(7)\)

\(\displaystyle LK=\frac{21}{5}\)

\(LK=4.2\)

\(\boxed{LK=4.2\text{ cm}}\)

Matching Corresponding Vertices and Sides

Corresponding Vertices

\(A\leftrightarrow G\)

\(B\leftrightarrow H\)

\(C\leftrightarrow I\)

\(D\leftrightarrow J\)

\(E\leftrightarrow K\)

\(F\leftrightarrow L\)

Corresponding Sides Used

\(\overline{AB}\leftrightarrow\overline{GH}\)

\(\overline{DE}\leftrightarrow\overline{JK}\)

\(\overline{FE}\leftrightarrow\overline{LK}\)

Matching the vertices in order prevents unrelated sides from being compared.

Another Method: Solve Proportions

Find \(\overline{JK}\)

Compare smaller to larger:

\(\displaystyle \frac{6}{10}=\frac{JK}{3}\)

Cross multiply:

\(10(JK)=18\)

\(\displaystyle JK=\frac{18}{10}\)

\(\boxed{JK=1.8\text{ cm}}\)

Find \(\overline{LK}\)

Use the same smaller-to-larger comparison:

\(\displaystyle \frac{6}{10}=\frac{LK}{7}\)

Cross multiply:

\(10(LK)=42\)

\(\displaystyle LK=\frac{42}{10}\)

\(\boxed{LK=4.2\text{ cm}}\)

Verify the Results

Each smaller-to-larger ratio should equal the original scale factor \(\frac{3}{5}\).

Known Pair

\(\displaystyle \frac{GH}{AB}=\frac{6}{10}\)

\(\displaystyle \frac{6}{10}=\frac{3}{5}\)

Bottom Sides

\(\displaystyle \frac{JK}{DE}=\frac{1.8}{3}\)

\(\displaystyle \frac{1.8}{3}=\frac{3}{5}\)

Right Sides

\(\displaystyle \frac{LK}{FE}=\frac{4.2}{7}\)

\(\displaystyle \frac{4.2}{7}=\frac{3}{5}\)

\(\boxed{JK=1.8\text{ cm}\quad\text{and}\quad LK=4.2\text{ cm}}\)

🧠 Math Vocabulary

• Similar figures: Figures with congruent corresponding angles and proportional corresponding side lengths.
• Similarity: The relationship between figures that have the same shape but may have different sizes.
• Corresponding vertices: Vertices that occupy matching positions in two figures.
• Corresponding sides: Sides that occupy matching positions in two figures.
• Ratio: A comparison of two quantities using division.
• Equivalent ratios: Ratios that represent the same comparison.
• Proportion: An equation showing that two ratios are equivalent.
• Scale factor: The constant multiplier used to enlarge or reduce a figure.
• Reduction: A scaled copy created with a scale factor between \(0\) and \(1\).
• Enlargement: A scaled copy created with a scale factor greater than \(1\).
• Cross products: Products formed by multiplying diagonally across a proportion.
• Missing side length: An unknown measurement found using proportional corresponding sides.
• Scaled copy: A figure produced by multiplying all lengths of another figure by the same scale factor.
• Constant of proportionality: The constant multiplier relating corresponding measurements.

💡 Main Idea

The side \(\overline{AB}\) of the larger figure corresponds to \(\overline{GH}\) of the smaller figure. Since \(AB=10\) centimeters and \(GH=6\) centimeters, the scale factor from the larger figure to the smaller figure is \(\frac{6}{10}=\frac{3}{5}\). Applying this same scale factor gives \(JK=\frac{3}{5}(3)=1.8\) centimeters and \(LK=\frac{3}{5}(7)=4.2\) centimeters.

📚 What You Should Already Know

Students should know how to identify corresponding parts of similar figures, write and simplify ratios, multiply fractions and decimals, solve proportions, and use cross products to find an unknown value.

🚀 What Comes Next

Students can extend this reasoning by solving multistep missing-side problems, finding unknown scale factors, comparing perimeter and area in similar figures, drawing scaled copies, and applying similarity to maps, models, and indirect measurement.

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