When the similarity ratio is m:n, the area ratio is m²:n².
📘 Example ①
Two similar triangles have a similarity ratio of 3:4. If the area of the smaller triangle is 18cm², find the area of the larger triangle.
Answer: Area ratio = 3²:4² = 9:16. Larger area = 18 × (16/9) = 32cm²
Two similar triangles have a similarity ratio of 3:4. If the area of the smaller triangle is 18cm², find the area of the larger triangle.
Answer: Area ratio = 3²:4² = 9:16. Larger area = 18 × (16/9) = 32cm²
📘 Example ②
In △ABC, DE ∥ BC and AD:AB=1:3. Find the area ratio of △ADE and △ABC.
Answer: Similarity ratio = AD:AB = 1:3. Area ratio = 1²:3² = 1:9
In △ABC, DE ∥ BC and AD:AB=1:3. Find the area ratio of △ADE and △ABC.
Answer: Similarity ratio = AD:AB = 1:3. Area ratio = 1²:3² = 1:9
💡 Key Point
- Similarity ratio m:n → area ratio m²:n²
- If the area ratio is known, the similarity ratio can be found by taking the square root.
Practice Problems
- Two figures have a similarity ratio of 5:2. If the area of the larger figure is 75cm², find the area of the smaller figure.
Answer and Explanation
- Answer: 12cm². Area ratio=25:4. Smaller area=75×(4/25)=12.