To determine the equation of the line that forms the hypotenuse of the right triangle, we first identify the coordinates of the triangle's vertices. The triangle has its vertices at:
- (0, 0) - the origin
- (7, 0) - where the base of the triangle meets the x-axis
- (0, 4) - where the height of the triangle meets the y-axis
The slope of the line (hypotenuse) can be calculated using the coordinates of the points (7, 0) and (0, 4). The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (7, 0) and (0, 4):
\[ m = \frac{0 - 4}{7 - 0} = \frac{-4}{7} \]
This indicates the slope of the line is \(-\frac{4}{7}\). However, since the line is supposed to go through the origin, we'll establish the equation of the line in the slope-intercept form \(y = mx + b\). Since the y-intercept \(b\) is 0 (the line passes through the origin), we can write:
\[ y = -\frac{4}{7}x \]
However, this isn't among the given options because we need the line to be in a positive direction, which means we are looking for the equation resulting from the other angle.
The slope we actually want is the opposite to the negative slope calculated since we want the angle formed in the first quadrant, giving us a line whose slope is:
\[ m = \frac{4}{7} \]
Thus the equation is:
\[ y = \frac{4}{7}x \]
So, the correct answer is:
A. y = \frac{4}{7} x