Similar triangles problems with step-by-step solutions Grade 8

📚 What are Similar Triangles?

Similar triangles are triangles that have the same shape but can be different sizes. This means their corresponding angles are equal, and their corresponding sides are in proportion. Understanding this relationship is key to solving problems involving similar triangles.

📜 A Little History

The concept of similar triangles dates back to ancient Greece, with mathematicians like Thales and Pythagoras using these principles to measure heights and distances. The formal study of similarity is a cornerstone of Euclidean geometry.

📐 Key Principles of Similar Triangles

• 📏 Angle-Angle (AA) Similarity: If two angles of one triangle are congruent (equal) to two angles of another triangle, then the triangles are similar.
• साइड Side-Side-Side (SSS) Similarity: If all three pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.
• साइड Side-Angle-Side (SAS) Similarity: If two pairs of corresponding sides of two triangles are in proportion, and the included angles are congruent, then the triangles are similar.

📝 Example Problem 1: Using AA Similarity

Triangle ABC has angles $\angle A = 60^\circ$ and $\angle B = 80^\circ$. Triangle XYZ has angles $\angle X = 60^\circ$ and $\angle Y = 80^\circ$. Are these triangles similar?

Solution:

Yes, by AA similarity, since two angles in $\triangle ABC$ are equal to two angles in $\triangle XYZ$.

📝 Example Problem 2: Using SSS Similarity

Triangle PQR has sides PQ = 3, QR = 4, and RP = 5. Triangle LMN has sides LM = 6, MN = 8, and NL = 10. Are these triangles similar?

Solution:

$\frac{PQ}{LM} = \frac{3}{6} = \frac{1}{2}$, $\frac{QR}{MN} = \frac{4}{8} = \frac{1}{2}$, and $\frac{RP}{NL} = \frac{5}{10} = \frac{1}{2}$. Since all corresponding sides are in the same proportion, $\triangle PQR \sim \triangle LMN$ by SSS similarity.

📝 Example Problem 3: Using SAS Similarity

In triangle DEF, DE = 4, DF = 6, and $\angle D = 45^\circ$. In triangle UVW, UV = 8, UW = 12, and $\angle U = 45^\circ$. Are these triangles similar?

Solution:

$\frac{DE}{UV} = \frac{4}{8} = \frac{1}{2}$ and $\frac{DF}{UW} = \frac{6}{12} = \frac{1}{2}$. Since $\angle D = \angle U$, $\triangle DEF \sim \triangle UVW$ by SAS similarity.

🌍 Real-World Applications

• 🗺️ Mapmaking: Cartographers use similar triangles to create scaled-down versions of geographical areas.
• 🏗️ Architecture: Architects apply similar triangles in designing structures and ensuring proportional accuracy.
• 📸 Photography: Photographers use principles of similar triangles to understand perspective and depth of field.

💡 Tips and Tricks

• ✅ Always label your triangles: Clearly label all vertices and sides to avoid confusion.
• 🔍 Look for shared angles: Shared angles often indicate similarity.
• ✍️ Write out the ratios: Expressing side ratios explicitly helps in verifying proportionality.

📝 Practice Quiz

1. ❓Triangle ABC has sides AB = 5, BC = 7, CA = 9. Triangle DEF has sides DE = 10, EF = 14, FD = 18. Are these triangles similar? Why or why not?
2. ❓Triangle GHI has $\angle G = 50^\circ$ and $\angle H = 70^\circ$. Triangle JKL has $\angle J = 50^\circ$ and $\angle K = 70^\circ$. Are these triangles similar? State the theorem.
3. ❓In triangle MNO, MN = 6, MO = 8, and $\angle M = 60^\circ$. In triangle PQR, PQ = 12, PR = 16, and $\angle P = 60^\circ$. Are these triangles similar? Show your work.

🔑 Conclusion

Understanding similar triangles is a fundamental skill in geometry with numerous practical applications. By mastering the AA, SSS, and SAS similarity criteria, you can confidently tackle a wide range of problems. Keep practicing, and you'll become a pro in no time!