Dilations & Similarity - Pre Algebra
- Circle A has radius 8 units and circle B has radius 12 units. If circle A is dilated by scale factor m and circle B is dilated by scale factor n, and both resulting circles are congruent, which equation correctly relates m and n? Answer: 8m = 12n For the dilated circles to be congruent, they must have equal radii: 8m = 12n. This means the final radius of circle A after dilation equals the final radius of circle B after dilation. Choice B incorrectly adds the expressions. Choice C relates areas instead of radii. Choice D gives the wrong ratio relationship.
- Triangle ABC is dilated by a scale factor of to create triangle A'B'C'. If the perimeter of triangle ABC is 28 units and the area is 42 square units, what is the area of triangle A'B'C'? Answer: 23.625 square units When a figure is dilated by scale factor k, the area is multiplied by k². Here, k = 3/4, so k² = 9/16. The new area is 42 × 9/16 = 23.625 square units. Choice A incorrectly multiplies perimeter by scale factor. Choice C incorrectly divides original area by 2. Choice D incorrectly multiplies area by scale factor instead of scale factor squared.
- Two similar triangles have areas in the ratio 16:25. If a dilation with scale factor k transforms the smaller triangle so that it becomes congruent to the larger triangle, what is the value of k? Answer: If area ratio is 16:25, then the ratio of corresponding sides is √16:√25 = 4:5. To make the smaller triangle congruent to the larger, we need scale factor k = 5/4. Choice A gives the ratio of smaller to larger sides. Choice C incorrectly uses (5-4)/16. Choice D uses the area ratio instead of side ratio.