Two similar triangles have a ratio of corresponding sides 3 : 4. Find the ratio of their areas.

Introduction / Context:
For similar figures, linear dimensions scale by a factor k, areas scale by k^2, and volumes (if any) scale by k^3. For triangles in particular, if the ratio of corresponding sides is known, the area ratio is simply the square of that linear ratio. This property follows from area formulas whose units are squared lengths.

Given Data / Assumptions:

• Triangles are similar; side ratio (triangle 1 : triangle 2) = 3 : 4.
• Thus the linear scale factor k = 3/4 from the smaller to the larger.

Concept / Approach:
Area ratio for similar figures is (side ratio)^2. Hence, the area ratio of the first to the second triangle equals (3/4)^2 = 9/16.

Step-by-Step Solution:
Let corresponding sides be 3x and 4x.Then areas are proportional to (3x)^2 and (4x)^2.Area ratio = 9x^2 : 16x^2 = 9 : 16.

Verification / Alternative check:
Pick concrete numbers: if sides are 3 and 4 with included equal angle, areas scale by base*height; both base and height scale by the same factor, giving a squared effect and confirming 9 : 16.

Why Other Options Are Wrong:
4 : 3 and 3 : 4 are linear ratios, not area ratios; √3 : 2 is irrelevant.

Common Pitfalls:
Confusing linear scaling (k) with area scaling (k^2). Always square the side ratio to get the area ratio for similar plane figures.

Final Answer:
9 : 16