Determining The Scale Factor Of Similar Figures

💡 Main Idea

When comparing similar figures, the scale factor depends on the direction of the comparison. Moving from a larger figure to a smaller figure gives a scale factor less than \(1\), while moving from a smaller figure to a larger figure gives a scale factor greater than \(1\). These two scale factors are reciprocals because they describe opposite directions.

✏️ Worked Examples

Problem from the video: If \(\triangle A \sim \triangle B\), determine the scale factor from \(A\) to \(B\) and from \(B\) to \(A\).

Triangle \(A\) has corresponding side lengths \(4\) cm and \(8\) cm. Triangle \(B\) has corresponding side lengths \(3\) cm and \(6\) cm.

Scale Factor from \(A\) to \(B\)

Use:

\(\text{Scale Factor}=\frac{\text{new shape}}{\text{original shape}}\)

From \(A\) to \(B\), triangle \(A\) is the original and triangle \(B\) is the new shape.

\(\frac{3}{4}=0.75\)

Check with the bases:

\(\frac{6}{8}=0.75\)

Scale factor from \(A\) to \(B\): \(\frac{3}{4}\) or \(0.75\). Since this is less than \(1\), triangle \(B\) is a reduction of triangle \(A\).

Scale Factor from \(B\) to \(A\)

Now triangle \(B\) is the original and triangle \(A\) is the new shape.

\(\frac{4}{3}=1.\overline{3}\)

Check with the bases:

\(\frac{8}{6}=\frac{4}{3}=1.\overline{3}\)

The scale factor is greater than \(1\), so the figure is enlarged.

Scale factor from \(B\) to \(A\): \(\frac{4}{3}\) or \(1.\overline{3}\). This is the reciprocal of \(\frac{3}{4}\).

Aha Moment

The scale factor changes depending on the direction. From \(A\) to \(B\), the triangle gets smaller, so the factor is \(\frac{3}{4}\). From \(B\) to \(A\), the triangle gets larger, so the factor is \(\frac{4}{3}\). These are reciprocals because they undo each other.

🧠 Math Vocabulary

• Similar Figures: Figures that have the same shape but may be different sizes.
• Corresponding Sides: Sides in similar figures that match each other based on position.
• Scale Factor: The multiplier that compares the dimensions of a new shape to the dimensions of the original shape.
• Reduction: A copy of a figure that is smaller than the original because the scale factor is less than \(1\).
• Enlargement: A copy of a figure that is larger than the original because the scale factor is greater than \(1\).
• Congruent Figures: Figures that have the same shape and the same size; this happens when the scale factor is \(1\).
• Reciprocal: A number formed by switching the numerator and denominator of a fraction.

📚 What You Should Already Know

Students should know how to identify corresponding sides, write ratios, simplify fractions, convert fractions to decimals, and understand that similar figures have proportional side lengths.

🚀 What Comes Next

After finding scale factors between similar figures, students can use scale factors to solve for missing side lengths, compare perimeter and area changes, and solve scale drawing and similarity word problems.

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