A line has a slope of $-\frac{4}{5}$. Which ordered pairs could be points on a line that is perpendicular to this line? Select two options.
A. ( $-2,0$ ) and ( 2,5 )
B. $(-4,5)$ and $(4,-5)$
C. $(-3,4)$ and $(2,0)$
D. $(1,-1)$ and $(6,-5)$
E. $(2,-1)$ and $(10,9)
Answer (2)
Calculate the slope between each pair of points using the formula m = x 2 − x 1 y 2 − y 1 .
Determine the slope of the perpendicular line, which is the negative reciprocal of − 5 4 , i.e., 4 5 .
Identify the pairs of points with a slope of 4 5 .
The pairs are ( − 2 , 0 ) and ( 2 , 5 ) , and ( 2 , − 1 ) and ( 10 , 9 ) .
( − 2 , 0 ) and ( 2 , 5 ) , ( 2 , − 1 ) and ( 10 , 9 )
Explanation
Understanding the Problem The problem asks us to identify two pairs of points that lie on a line perpendicular to a line with a slope of − 5 4 . The slope of a line perpendicular to a line with slope m is the negative reciprocal of m . Therefore, the slope of the perpendicular line is 4 5 . We need to calculate the slope between each pair of points and check if it is equal to 4 5 .
Calculating Slopes The slope between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula: m = x 2 − x 1 y 2 − y 1 We will calculate the slope for each of the given pairs of points.
Slope Calculation for Each Pair
Pair 1: ( − 2 , 0 ) and ( 2 , 5 ) m = 2 − ( − 2 ) 5 − 0 = 4 5 = 1.25
Pair 2: ( − 4 , 5 ) and ( 4 , − 5 ) m = 4 − ( − 4 ) − 5 − 5 = 8 − 10 = − 4 5 = − 1.25
Pair 3: ( − 3 , 4 ) and ( 2 , 0 ) m = 2 − ( − 3 ) 0 − 4 = 5 − 4 = − 0.8
Pair 4: ( 1 , − 1 ) and ( 6 , − 5 ) m = 6 − 1 − 5 − ( − 1 ) = 5 − 4 = − 0.8
Pair 5: ( 2 , − 1 ) and ( 10 , 9 ) m = 10 − 2 9 − ( − 1 ) = 8 10 = 4 5 = 1.25
Identifying Perpendicular Pairs We are looking for pairs with a slope of 4 5 . From the calculations above, we see that:
Pair 1 has a slope of 4 5 .
Pair 5 has a slope of 4 5 .
Therefore, the ordered pairs that could be points on a line that is perpendicular to the given line are ( − 2 , 0 ) and ( 2 , 5 ) , and ( 2 , − 1 ) and ( 10 , 9 ) .
Final Answer The two pairs of points that lie on a line perpendicular to the line with slope − 5 4 are:
( − 2 , 0 ) and ( 2 , 5 )
( 2 , − 1 ) and ( 10 , 9 )
Examples
Understanding perpendicular slopes is crucial in various real-world applications. For instance, architects use this concept to design buildings with walls that meet at right angles, ensuring structural stability. Similarly, in navigation, understanding perpendicular directions helps in plotting efficient routes and avoiding obstacles. In computer graphics, perpendicularity is essential for creating realistic lighting and shadows, enhancing the visual appeal of virtual environments. These examples highlight how a solid grasp of perpendicular slopes is vital in diverse fields, contributing to precision and functionality.
The points that lie on lines perpendicular to a line with a slope of − 5 4 correspond to pairs A and E. Pair A includes the points ( − 2 , 0 ) and ( 2 , 5 ) , while pair E includes the points ( 2 , − 1 ) and ( 10 , 9 ) , both with a slope of 4 5 . Therefore, the answer is A and E.
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