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 (1)
Calculate the slope of the line containing (3,4) and (-2,2): m = − 2 − 3 2 − 4 = 0.4 .
Calculate the slope of the line containing (-2,-5) and (-7,-3): m = − 7 − ( − 2 ) − 3 − ( − 5 ) = − 0.4 .
Calculate the slope of the line containing (-1,1) and (-6,-1): m = − 6 − ( − 1 ) − 1 − 1 = 0.4 .
Calculate the slope of the line containing (0,0) and (2,5): m = 2 − 0 5 − 0 = 2.5 .
Calculate the slope of the line containing (1,0) and (6,2): m = 6 − 1 2 − 0 = 0.4 .
Calculate the slope of the line containing (3,0) and (8,2): m = 8 − 3 2 − 0 = 0.4 .
The pairs of points that have the same slope as the line containing (3,4) and (-2,2) are the correct answers: ( − 1 , 1 ) and ( − 6 , − 1 ) , ( 1 , 0 ) and ( 6 , 2 ) , ( 3 , 0 ) and ( 8 , 2 ) .
Explanation
Understanding the Problem We are given two points ( 3 , 4 ) and ( − 2 , 2 ) that define a line. We need to find which of the given pairs of points lie on a line parallel to the line defined by ( 3 , 4 ) and ( − 2 , 2 ) . Parallel lines have the same slope. The slope of a 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 .
Calculating the Slope of the Given Line First, let's calculate the slope of the line containing the points ( 3 , 4 ) and ( − 2 , 2 ) . Using the slope formula, we have: m = − 2 − 3 2 − 4 = − 5 − 2 = 5 2 = 0.4
Checking the Slopes of the Other Lines Now, we need to calculate the slope for each of the given pairs of points and check if it is equal to 0.4 .
Analyzing the First Pair of Points
For the points ( − 2 , − 5 ) and ( − 7 , − 3 ) , the slope is: m = − 7 − ( − 2 ) − 3 − ( − 5 ) = − 5 2 = − 0.4 This slope is not equal to 0.4 , so this pair of points does not lie on a parallel line.
Analyzing the Second Pair of Points
For the points ( − 1 , 1 ) and ( − 6 , − 1 ) , the slope is: m = − 6 − ( − 1 ) − 1 − 1 = − 5 − 2 = 5 2 = 0.4 This slope is equal to 0.4 , so this pair of points could lie on a parallel line.
Analyzing the Third Pair of Points
For the points ( 0 , 0 ) and ( 2 , 5 ) , the slope is: m = 2 − 0 5 − 0 = 2 5 = 2.5 This slope is not equal to 0.4 , so this pair of points does not lie on a parallel line.
Analyzing the Fourth Pair of Points
For the points ( 1 , 0 ) and ( 6 , 2 ) , the slope is: m = 6 − 1 2 − 0 = 5 2 = 0.4 This slope is equal to 0.4 , so this pair of points could lie on a parallel line.
Analyzing the Fifth Pair of Points
For the points ( 3 , 0 ) and ( 8 , 2 ) , the slope is: m = 8 − 3 2 − 0 = 5 2 = 0.4 This slope is equal to 0.4 , so this pair of points could lie on a parallel line.
Final Answer Therefore, the ordered pairs that could be points on a line parallel to the line that contains ( 3 , 4 ) and ( − 2 , 2 ) are: ( − 1 , 1 ) and ( − 6 , − 1 ) ( 1 , 0 ) and ( 6 , 2 ) ( 3 , 0 ) and ( 8 , 2 )
Examples
Understanding parallel lines is crucial in architecture and construction. When designing buildings, architects ensure that walls and support beams are parallel to maintain structural integrity and aesthetic appeal. For example, if a wall is not perfectly parallel to the foundation, it can lead to uneven weight distribution and potential structural failure. The concept of parallel lines and their slopes helps architects create precise and stable designs.
