To determine if the height of a right triangle is a function of its area, we need to analyze the relationship between area and height.
The area \( A \) of a right triangle can be expressed with the formula:
\[ A = \frac{1}{2} \times b \times h \]
where \( b \) is the base length of the triangle and \( h \) is the height of the triangle.
If we are given the area \( A \) and want to find the height \( h \), we can manipulate the formula:
\[ h = \frac{2A}{b} \]
From this equation, we can see that for each area \( A \), the height \( h \) depends on the value of the base \( b \). If the base \( b \) is fixed, then for a given area, there is one unique height. However, if \( b \) can vary, then for the same area \( A \), there may be different possible heights \( h \) depending on the chosen base \( b \).
This means that the height of a right triangle can have multiple outputs (heights) for the same input (area), depending on the base length used.
Therefore, we conclude that:
The height of a right triangle is not a function of its area because for each input (area), there are multiple possible outputs (heights) depending on the choice of the base length \( b \).
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