In a triangle, each of the three sides is reduced to half of its original length. By what percentage does the area of the triangle decrease as a result of halving all the sides?

Introduction / Context:
This aptitude question checks understanding of how the area of a triangle changes when all side lengths are scaled by a common factor. Instead of working with specific numbers, we look at the relationship between linear dimensions and area. Questions like this are common in geometry and mensuration, especially in exams where students must quickly see how scaling affects area or volume without recomputing from scratch.

Given Data / Assumptions:
- We have an original triangle with some base and height or with some sides. - Each side of the triangle is halved, so linear dimensions are scaled by a factor of 1/2. - We are asked to find the percentage decrease in area, not the new area itself. - The shape remains a similar triangle after scaling, only the size changes.

Concept / Approach:
The key concept is similarity and scaling. When all the sides of a triangle are multiplied by a factor k, the area is multiplied by k^2. Here, the factor is 1/2. Therefore the new area becomes (1/2)^2 times the original area. Once we know the ratio of new area to old area, we can easily compute the percentage decrease by comparing the old area and the new area and expressing the reduction as a percentage of the original area.

Step-by-Step Solution:
Step 1: Let the original area of the triangle be A square units. Step 2: Each side is halved. That means every linear dimension of the triangle is multiplied by 1/2. Step 3: For similar figures, area scales with the square of the linear scale factor. So new area = A * (1/2)^2. Step 4: Compute (1/2)^2 = 1/4. Step 5: Therefore the new area is A / 4, which is one fourth of the original area. Step 6: The amount of decrease in area is original area minus new area which is A - A / 4 = 3A / 4. Step 7: To find the percentage decrease, divide the decrease by the original area and multiply by 100 percent. Step 8: Percentage decrease = (3A / 4) / A * 100% = (3 / 4) * 100% = 75%.

Verification / Alternative check:
Take a simple example for verification. Suppose a right triangle has base 4 units and height 6 units. Its area is (1/2) * 4 * 6 = 12 square units. Now halve both base and height: new base = 2 units and new height = 3 units, so new area = (1/2) * 2 * 3 = 3 square units. Compare: original area 12, new area 3. The decrease is 9 square units. Compute percentage decrease: 9 / 12 * 100% = 75%. This matches the general derivation.

Why Other Options Are Wrong:
Option 50% would be correct if area depended linearly on side length, but area depends on the square of the scale factor, not the first power. Option 25% confuses the remaining area with the decrease. The new area is 25% of the original, not the decrease. Option No change is incorrect because the triangle clearly becomes smaller when sides are halved, so the area must reduce.

Common Pitfalls:
Many learners treat area as if it scales in direct proportion to side length rather than with the square of the side length. Another mistake is mixing up the remaining percentage and the percentage decrease. Here the new area is 25% of the original, so the percentage decrease is 75%, not 25%. Always carefully distinguish between what remains and what is lost. Remember that scaling laws for area and volume are powerful shortcuts for many exam problems.

Final Answer:
When each side of the triangle is halved, the area decreases by 75%.