Which equation represents a line that is perpendicular to the line passing through $(-4,7)$ and $(1,3)$?
A. $y=-\frac{5}{4} x-2$
B. $y=\frac{5}{4} x+8$
C. $y=\frac{4}{5} x-3$
D. $y=-\frac{4}{5} x+6$
Answer (2)
Calculate the slope of the line passing through ( − 4 , 7 ) and ( 1 , 3 ) using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = − 5 4 .
Determine the slope of the perpendicular line by taking the negative reciprocal of the original slope: m ⊥ = 4 5 .
Compare the calculated perpendicular slope with the slopes of the lines given in the options.
Identify the equation with the correct slope, which is y = 4 5 x + 8 .
B
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 follows: m = x 2 − x 1 y 2 − y 1 Substituting the given points: m = 1 − ( − 4 ) 3 − 7 = 5 − 4 = − 5 4 So, the slope of the line 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 m ⊥ is: m ⊥ = − m 1 In our case, m = − 5 4 , so: m ⊥ = − − 5 4 1 = 4 5 Thus, the slope of the perpendicular line is 4 5 .
Identify the correct equation. Now, we need to find the equation of the line that has a slope of 4 5 . The equation is in the form y = m x + b , where m is the slope. We are looking for an equation with m = 4 5 .
Looking at the options: A. y = − 4 5 x − 2 (slope is − 4 5 )
B. y = 4 5 x + 8 (slope is 4 5 )
C. y = 5 4 x − 3 (slope is 5 4 )
D. y = − 5 4 x + 6 (slope is − 5 4 )
Option B has the correct slope of 4 5 .
State the 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 So, the correct answer is B.
Examples
Understanding perpendicular lines is crucial in architecture and construction. For instance, when designing a building, ensuring walls are perpendicular to the ground is essential for stability. If a surveyor determines a line has a slope of − 5 4 , an architect would use the negative reciprocal, 4 5 , to ensure the supporting walls are perfectly perpendicular, preventing structural issues.
The line that is perpendicular to the line passing through ( − 4 , 7 ) and ( 1 , 3 ) has a slope of 4 5 . The correct equation representing this line is option B: y = 4 5 x + 8 . Hence, the answer to the question is B.
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