To find the equation of a line that is perpendicular to the line defined by the points (0, 4) and (-8, 0) and shares the same y-intercept, we need to follow these steps:
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Find the slope of the original line: The slope (m) of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as follows:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting in the values of the points (0, 4) and (-8, 0):
\[ m = \frac{0 - 4}{-8 - 0} = \frac{-4}{-8} = \frac{1}{2} \]
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Find the slope of the perpendicular line: The slope of a line that is perpendicular to another can be found by taking the negative reciprocal of the original slope. Thus, if the slope of the original line is \(\frac{1}{2}\), the slope of the perpendicular line will be:
\[ m_{perpendicular} = -\frac{1}{(\frac{1}{2})} = -2 \]
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Form the equation of the perpendicular line: Since the y-intercept is shared with the original line, and given that the original line has a y-intercept of \(4\) at \((0, 4)\), the equation of the perpendicular line can be written in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Substituting \(m = -2\) and \(b = 4\):
\[ y = -2x + 4 \]
Now, let's compare this equation with the options given:
a. \(y = -\frac{1}{2}x + 4\)
b. \(y = 2x - 8\)
c. \(y = -2x + 4\)
d. \(y = \frac{1}{2}x - 8\)
The correct answer is:
c. \(y = -2x + 4\)
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