Given: AB || CD. If the coordinates of point A are (8,0) and the coordinates of point B are (3,7), what is the y-intercept of AB? If the coordinates of point D are (5, 5), the equation of line CD is y= x+
Answer (2)
Calculate the slope of line AB using points A(8, 0) and B(3, 7): m = 3 − 8 7 − 0 = − 5 7 = − 1.4 .
Determine the equation of line AB using point-slope form and convert to slope-intercept form to find the y-intercept: y = − 5 7 x + 5 56 , so the y-intercept is 11.2.
Since AB || CD, line CD has the same slope: m = − 5 7 .
Find the equation of line CD using point D(5, 5) and the slope m = − 5 7 : y = − 5 7 x + 12 .
y = − 5 7 x + 12
Explanation
Calculate the slope of line AB First, we need to find the slope of line AB using the coordinates of points A(8, 0) and B(3, 7). The slope, denoted as 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 coordinates of A and B: m = 3 − 8 7 − 0 = − 5 7 = − 5 7 = − 1.4 So, the slope of line AB is -1.4.
Find the equation of line AB and its y-intercept Next, we use the point-slope form of a line to find the equation of line AB. The point-slope form is given by: y − y 1 = m ( x − x 1 ) Using point A(8, 0) and the slope m = − 5 7 , we have: y − 0 = − 5 7 ( x − 8 ) y = − 5 7 x + 5 56 To find the y-intercept, we set x = 0 :
y = − 5 7 ( 0 ) + 5 56 = 5 56 = 11.2 Thus, the y-intercept of line AB is 11.2.
Find the equation of line CD Since line CD is parallel to line AB, it has the same slope. Therefore, the slope of line CD is also m = − 5 7 = − 1.4 . We use the point-slope form again, this time with point D(5, 5): y − 5 = − 5 7 ( x − 5 ) y − 5 = − 5 7 x + 7 y = − 5 7 x + 7 + 5 y = − 5 7 x + 12 So, the equation of line CD is y = − 5 7 x + 12 .
Final Answer Therefore, the y-intercept of AB is 11.2, and the equation of line CD is y = − 5 7 x + 12 .
Examples
Understanding linear equations and parallel lines is crucial in various real-world applications. For instance, consider city planning where streets are designed to be parallel to each other. If you know the equation of one street (line), you can determine the equation of another parallel street using the principles we applied here. This ensures efficient and organized urban layouts, optimizing traffic flow and infrastructure planning.
The y-intercept of line AB is 11.2, calculated from the slope of the line derived from points A(8, 0) and B(3, 7). The equation of the parallel line CD, using point D(5, 5) and the same slope, is y = -7/5x + 12.
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