A square of side x has the same area as a triangle whose base is x. Find the altitude (height) of the triangle in terms of x.

Introduction / Context:
This is a direct comparison between the area of a square and the area of a triangle. Setting the two areas equal and solving for the triangle’s altitude in terms of the shared base x gives a quick algebraic result. It illustrates how linear dimensions relate when areas are equated across different shapes.

Given Data / Assumptions:

• Square side = x ⇒ A_square = x^2
• Triangle base = x; altitude = h
• Triangle area A_triangle = (1/2) * base * height = (1/2) * x * h

Concept / Approach:
Equate x^2 with (1/2) * x * h and solve for h. Since x > 0 for a geometric length, divide safely by x to isolate h.

Step-by-Step Solution:
x^2 = (1/2) * x * hDivide both sides by x (x > 0): x = (1/2) * hh = 2x

Verification / Alternative check:
Plug h = 2x back: triangle area = 0.5 * x * 2x = x^2, equal to the square’s area as required.

Why Other Options Are Wrong:
x/2 and x yield areas smaller than x^2; 4x yields an area of 2x^2, which would be too large. Only 2x balances the two areas exactly.

Common Pitfalls:
Forgetting the 1/2 factor in the triangle area formula or canceling x incorrectly can lead to wrong multiples of x for the altitude.

Final Answer:
2x