Which equation represents a line that is perpendicular to the line passing through $(-4,7)$ and $(1,3)$?
A. $y=\frac{4}{5} x-3$
B. $y=-\frac{5}{4} x-2$
C. $y=\frac{5}{4} x+8$
D. $y=-\frac{4}{5} x+6$
Answer (1)
Calculate the slope of the line passing through ( − 4 , 7 ) and ( 1 , 3 ) : m = 1 − ( − 4 ) 3 − 7 = − 5 4 .
Determine the slope of the perpendicular line: m ⊥ = − m 1 = 4 5 .
Identify the equation with the slope 4 5 among the given options.
The equation representing the perpendicular line is: y = 4 5 x + 8 .
Explanation
Calculate the slope of the given line. First, we need to find the slope of the line passing through the points ( − 4 , 7 ) and ( 1 , 3 ) . The slope, m , is calculated as the change in y divided by the change in x : m = x 2 − x 1 y 2 − y 1 Substituting the given points, we get: m = 1 − ( − 4 ) 3 − 7 = 5 − 4 = − 5 4 So, the slope of the line passing through ( − 4 , 7 ) and ( 1 , 3 ) is − 5 4 .
Determine the slope of the perpendicular line. Next, we need to find the slope of a line perpendicular to the given line. The slope of a perpendicular line is the negative reciprocal of the original line's slope. If the original slope is m , the perpendicular slope is − m 1 .
In our case, the original slope is − 5 4 , so the perpendicular slope is: m ⊥ = − − 5 4 1 = 4 5 Thus, the slope of the perpendicular line is 4 5 .
Compare with the given options. Now, we need to identify which of the given options has a slope of 4 5 . The equation of a line in slope-intercept form is y = m x + b , where m is the slope and b is the y-intercept.
Let's examine the options:
A. y = 5 4 x − 3 has a slope of 5 4 .
B. y = − 4 5 x − 2 has a slope of − 4 5 .
C. y = 4 5 x + 8 has a slope of 4 5 .
D. y = − 5 4 x + 6 has a slope of − 5 4 .
Option C, y = 4 5 x + 8 , has the correct slope of 4 5 .
Final Answer. Therefore, the equation that represents a line perpendicular to the line passing through ( − 4 , 7 ) and ( 1 , 3 ) is y = 4 5 x + 8 .
Examples
Understanding perpendicular lines is crucial in many real-world applications, such as architecture and construction. For example, when building a house, the walls need to be perpendicular to the ground to ensure stability. If a surveyor determines the slope of the ground, they can calculate the slope needed for the walls to be perpendicular, ensuring the structure is sound and safe. This principle also applies to designing roads, bridges, and other infrastructure where angles and slopes are critical for functionality and safety.
