Triangle Similarity – Solved Examples

Table of Contents

• Example 1: Calculating the similarity ratio
• Example 2: Finding missing sides
• Example 3: Determining similarity using the SSS Theorem
• Example 4: Using the SAS Theorem
• Example 5: Using the AA Theorem
• Example 6: Word problem – height of a tree
• Related Articles

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Example 1: Calculating the similarity ratio

Problem: Triangle $ABC$ has sides $a=5$ cm, $b=8$ cm, $c=10$ cm. Triangle $DEF$ has sides $d=7.5$ cm, $e=12$ cm, $f=15$ cm. Find the similarity ratio. Solution:

We arrange the sides by length and compute the ratios:

All ratios are equal, so $△ABC\sim △DEF$ with a scale factor of $k=1.5$.

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Example 2: Finding missing sides

Problem: Triangles $ABC\sim DEF$ with a scale factor of $k=3$. The sides of $△ABC$ are: $a=4$ cm, $b=5$ cm, $c=7$ cm. Find the sides of $△DEF$. Solution:

Since $k=3$, we multiply each side by the scale factor:

The sides of $△DEF$ are: $d=12$ cm, $e=15$ cm, $f=21$ cm.

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Example 3: Determining similarity using the SSS Theorem

Problem: Are triangles $ABC$ ($a=6$, $b=9$, $c=12$) and $DEF$ ($d=4$, $e=6$, $f=8$) similar? Solution:

We arrange the sides by length:

$△ABC$: $6,9,12$ and $△DEF$: $4,6,8$

We compute the ratios of corresponding sides:

All ratios are equal ($k=1.5$), so yes, the triangles are similar by the SSS Theorem.

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Example 4: Using the SAS Theorem

Problem: $△ABC$: $a=5$ cm, $b=8$ cm, angle $\gamma =45°$. $△DEF$: $d=10$ cm, $e=16$ cm, angle ${\gamma }^{\prime }=45°$. Are they similar? Solution:

We check the ratios of two sides:

The ratios are equal ($k=2$) and the included angle is equal ($\gamma ={\gamma }^{\prime }=45°$).

By the SAS Theorem, the triangles are similar: $△ABC\sim △DEF$.

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Example 5: Using the AA Theorem

Problem: In triangle $ABC$, $\alpha =35°$ and $\beta =85°$. In triangle $DEF$, $\delta =35°$ and $\epsilon =85°$. Are the triangles similar? Solution:

Two angles match: $\alpha =\delta =35°$ and $\beta =\epsilon =85°$.

The third angle must also match:

By the AA Theorem, the triangles are similar: $△ABC\sim △DEF$.

💡 To apply the AA Theorem, it was enough to verify two angles — the third matches automatically.

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Example 6: Word problem – height of a tree

Problem: A pole $2$ m tall casts a shadow $3$ m long. At the same time, a tree casts a shadow $12$ m long. What is the height of the tree? Solution:

The sun's rays strike at the same angle, so the pole's shadow and the tree's shadow form similar right triangles (same angle of incidence + right angle = AA Theorem).

Scale factor:

Height of the tree:

The tree is 8 m tall.

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Related Articles

• Comprehensive guide: Triangle Similarity – Comprehensive Guide
• Similarity of figures: Similarity of Geometric Figures
• Similarity ratio: Similarity Ratio (Scale Factor)
• Similarity theorems: Triangle Similarity Theorems – SSS, SAS, AA

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Practice

• 🔢 Similarity ratio: Calculate the Similarity Ratio
• 📐 Missing sides: Find the Missing Side
• 🔍 Similarity theorems: Identify the Similarity Theorem
• 🧮 Mixed problems: Mixed Similarity Problems