Solve problems by combining knowledge of similarity ratios, area ratios, and parallel lines and ratios.
📘 Example ①
Find the area ratio of two triangles with a similarity ratio of 3:4.
Answer: Area ratio = square of the similarity ratio = 3²:4² = 9:16
Find the area ratio of two triangles with a similarity ratio of 3:4.
Answer: Area ratio = square of the similarity ratio = 3²:4² = 9:16
📘 Example ② (Height of a Tree)
A tree's shadow is 8m long. At the same time, a person 1.5m tall casts a 2m shadow. Find the height of the tree.
Answer: Tree:person = 8:2 = 4:1 (ratio of shadows). Height of tree = 1.5 × 4 = 6m
A tree's shadow is 8m long. At the same time, a person 1.5m tall casts a 2m shadow. Find the height of the tree.
Answer: Tree:person = 8:2 = 4:1 (ratio of shadows). Height of tree = 1.5 × 4 = 6m
💡 Key Point
- Similarity ratio m:n → area ratio m²:n²
- Similarity is also used in real life for shadows, scale drawings, and so on.
Practice Problems
- Two triangles have a similarity ratio of 2:5. If the area of the smaller triangle is 8cm², find the area of the larger triangle.
Answer and Explanation
- Answer: 50cm². Area ratio = 4:25. Larger area = 8×(25/4)=50.