A square and a rhombus stand on the same base and lie between the same two parallel lines. What is the ratio of the area of the square to the area of the rhombus in this situation?

Introduction:
This is a conceptual geometry question that tests a fundamental idea: when two plane figures stand on the same base and lie between the same parallels, they have the same height. Since area is equal to base multiplied by height for many standard shapes, this property often leads to simple area relationships. Here, we compare a square and a rhombus, both resting on the same base and between the same parallels, and we are asked for the ratio of their areas.

Given Data / Assumptions:

• There is a square standing on a given base segment. • There is a rhombus standing on the same base segment. • Both figures lie between the same pair of parallel lines, meaning they have equal perpendicular height. • We are asked to find the ratio: area of square : area of rhombus.

Concept / Approach:
The area of any parallelogram-like figure, including a rectangle, square, rhombus, or general parallelogram, is given by base * height, where height is the perpendicular distance between the pair of opposite sides. If two such figures share the same base and have the same height (that is, they lie between the same parallels), then their areas are equal. The shape of the figure may change the side lengths or angles, but as long as both base and height are common, the area remains the same. This is a well-known geometric result often used to compare parallelograms and related figures.

Step-by-Step Solution:
Step 1: Let the common base length be b units for both the square and the rhombus. Step 2: Let the common perpendicular distance between the parallels (height) be h units. Step 3: Area of the square in this configuration. Even though a square normally has all sides equal, in this conceptual setup we only care that it stands on base b and has height h. Therefore, its area = base * height = b * h. Step 4: Area of the rhombus in this configuration. A rhombus also has area = base * height, where base is any side and height is the perpendicular distance to the opposite side. Here, base = b and height = h, so area of the rhombus = b * h. Step 5: Compute the ratio of areas. Area of square : area of rhombus = (b * h) : (b * h) = 1 : 1.

Verification / Alternative check:
We can take any specific numeric example to verify the concept. Suppose the base is 10 units and the common height is 6 units. Then area of the square standing on that base and height is 10 * 6 = 60 square units. The rhombus also has area 10 * 6 = 60 square units. The ratio is again 60 : 60 = 1 : 1. No matter what actual values of base and height are chosen (as long as they are the same for both figures), the ratio stays equal to 1 : 1.

Why Other Options Are Wrong:
Option 1 : 2: This would mean the rhombus always has double the area of the square, which is not true when they share both base and height. Option 1 : 3 and Option 1 : 4: These suggest that the square has much smaller area than the rhombus with the same base and height, which contradicts the base * height formula. Option 2 : 1: This would mean the square always has twice the area of the rhombus, which is also false under the given conditions.

Common Pitfalls:
Many learners mistakenly focus on the fact that a square has all angles equal to 90° and a rhombus generally has oblique angles, and therefore assume their areas must differ. However, area depends on base and perpendicular height, not on angles directly. Another misconception is to assume that the side length defines the area in all configurations without considering the effective height. Remember, when two figures share the same base and lie between the same parallels, their areas are equal regardless of their shape along the top edge.

Final Answer:
The ratio of the area of the square to the area of the rhombus is 1 : 1.