If corresponding sides of two similar triangles are scaled by a factor k, what is the ratio of their perimeters?

• A. k^2
• B. 1/k
• C. sqrt(k)
• D. k

Answer: (D)

When two figures are similar, every linear measurement scales by the same factor. If you scale a triangle by k, each side length becomes k times as long, and the perimeter is just the sum of those three side lengths. Since all three sides increase by a factor of k, the entire perimeter also increases by k. So the second triangle’s perimeter is k times the first triangle’s perimeter, making the ratio of perimeters equal to k. For a quick check, doubling the sides doubles the perimeter, and scaling by 1/3 reduces the perimeter to one-third. For example, a triangle with sides 3, 4, 5 has perimeter 12; scaling by 2 gives sides 6, 8, 10 with perimeter 24, a ratio of 24 to 12 which is 2, matching k.