The area of a square with side a is equal to the area of a triangle with base a. What is the altitude (height) of the triangle?

Introduction / Context:This question tests equating areas of two shapes and solving for an unknown dimension. A square of side a has a fixed area a*a = a^2. A triangle with base a and altitude h has area (1/2)*a*h. If these two areas are equal, we set a^2 = (1/2)*a*h and solve for h. The key is careful cancellation of a (assuming a is not zero) and keeping the triangle area formula correct with the 1/2 factor. This is a straightforward substitution and algebra simplification problem commonly used in aptitude tests for quick concept checking.

Given Data / Assumptions:

• Square side = a
• Area of square = a^2
• Triangle base = a
• Triangle altitude = h
• Area of triangle = (1/2)*a*h

Concept / Approach:Equate the two areas and solve for h: a^2 = (1/2)*a*h. Cancel a and isolate h.

Step-by-Step Solution: Area(square) = a^2 Area(triangle) = (1/2)*a*h Given they are equal: a^2 = (1/2)*a*h Cancel a from both sides: a = (1/2)*h Multiply both sides by 2: h = 2a

Verification / Alternative check:If h = 2a, triangle area becomes (1/2)*a*(2a) = a^2, exactly matching the square’s area. This direct substitution confirms the result without extra steps.

Why Other Options Are Wrong: a would make triangle area (1/2)*a*a = a^2/2, only half of the square. 3a and 5a make triangle area larger than the square. a/2 makes triangle area far smaller than the square.

Common Pitfalls:Forgetting the 1/2 in triangle area, canceling incorrectly, or assuming altitude equals base without using the equality condition.

Final Answer:The altitude of the triangle is 2a.