Finding Missing Side Lengths with Similar Triangles

✏️ Worked Example

Problem From the Video: Find \(AC\)

Given:

\(AB = 16\text{ cm}\)

\(BD = 12\text{ cm}\)

\(DE = 9\text{ cm}\)

\(AC = ?\)

Step 1: Find the missing vertical side of the small triangle.

\(AD = AB - BD\)

\(AD = 16 - 12\)

\(AD = 4\text{ cm}\)

Step 2: Identify the similar triangles.

The small triangle \( \triangle BDE \) is similar to the large triangle \( \triangle BAC \).

Step 3: Match corresponding sides.

\(BD\) corresponds to \(BA\), and \(DE\) corresponds to \(AC\).

Step 4: Set up a proportion.

\( \frac{BD}{BA} = \frac{DE}{AC} \)

\( \frac{12}{16} = \frac{9}{AC} \)

Step 5: Simplify and solve.

\( \frac{12}{16} = \frac{3}{4} \)

\( \frac{3}{4} = \frac{9}{AC} \)

Since \(3 \times 3 = 9\), multiply the denominator by \(3\) also.

\(4 \times 3 = 12\)

Answer: \(AC = 12\text{ cm}\)

🧠 Math Vocabulary

• Similar Figures: figures with the same shape and proportional side lengths.
• Corresponding Sides: sides that match between similar figures.
• Scale Factor: the multiplier that compares corresponding side lengths.
• Proportion: an equation showing two ratios are equal.
• Triangle Inside a Triangle: a similarity setup where a smaller triangle is formed inside a larger triangle.

📚 What You Should Already Know

Students should already know how to write ratios, solve proportions, subtract side lengths, and identify corresponding sides in similar figures.

🚀 What Comes Next

After using similar triangles to find missing side lengths, students can apply the same proportional reasoning to scale drawings, indirect measurement, dilations, and real-world similarity problems.

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