Which ordered pairs could be points on a line parallel to the line that contains (3, 4) and (-2, 2)? Check all that apply.
A. (-2,-5) and (-7,-3)
B. (-1, 1) and (-6, -1)
C. (0,0) and (2,5)
D. (1, 0) and (6, 2)
E. (3,0) and (8, 2)
Answer (2)
Calculate the slope of the line passing through (3, 4) and (-2, 2), which is 5 2 = 0.4 .
Calculate the slopes of the lines passing through each of the given pairs of points.
Identify the pairs of points that have a slope of 0.4.
The pairs of points that lie on lines parallel to the given line are: (-1, 1) and (-6, -1); (1, 0) and (6, 2); (3, 0) and (8, 2).
Explanation
Problem Analysis We are given two points (3, 4) and (-2, 2) and asked to find which of the given pairs of points lie on a line parallel to the line containing these two points. Parallel lines have the same slope. Therefore, we need to calculate the slope of the line passing through (3, 4) and (-2, 2) and then check which of the other pairs of points have the same slope.
Calculating the Slope The slope of the line passing through points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula: m = x 2 − x 1 y 2 − y 1 For the points (3, 4) and (-2, 2), the slope is: m = − 2 − 3 2 − 4 = − 5 − 2 = 5 2 = 0.4
Checking Slopes of Given Pairs Now we need to calculate the slopes for each of the given pairs of points and check if they are equal to 0.4.
(-2, -5) and (-7, -3): m = − 7 − ( − 2 ) − 3 − ( − 5 ) = − 5 2 = − 0.4
(-1, 1) and (-6, -1): m = − 6 − ( − 1 ) − 1 − 1 = − 5 − 2 = 5 2 = 0.4
(0, 0) and (2, 5): m = 2 − 0 5 − 0 = 2 5 = 2.5
(1, 0) and (6, 2): m = 6 − 1 2 − 0 = 5 2 = 0.4
(3, 0) and (8, 2): m = 8 − 3 2 − 0 = 5 2 = 0.4
Identifying Parallel Lines Comparing the slopes, we find that the following pairs of points have a slope of 0.4, which is the same as the slope of the line passing through (3, 4) and (-2, 2):
(-1, 1) and (-6, -1)
(1, 0) and (6, 2)
(3, 0) and (8, 2)
Therefore, these pairs of points lie on lines parallel to the line containing (3, 4) and (-2, 2).
Examples
Understanding parallel lines is crucial in architecture and construction. For example, when designing a building, architects ensure that walls are parallel to each other for structural stability and aesthetic appeal. Similarly, in road construction, lanes are designed to be parallel to maintain a consistent flow of traffic and ensure safety. The concept of parallel lines and their slopes is also used in mapping and navigation systems to determine routes and directions.
The pairs of points that can represent lines parallel to the line containing (3, 4) and (-2, 2) are B: (-1, 1) and (-6, -1), D: (1, 0) and (6, 2), and E: (3, 0) and (8, 2). All these pairs share the same slope of 5 2 as the original line. Therefore, the correct options are B, D, and E.
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