Line through the points (-2, 6) and (4, 8) is perpendicular to the line through the points (8, 12) and (x, 24). Find the value of x.
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Step-by-Step Solution
Step 1: Calculate the slope of the first line
To find the slope of the first line, we use the given points (−2,6) and (4,8). The formula for the slope m between two points (x1,y1) and (x2,y2) is m=x2−x1y2−y1. Substituting the coordinates, we get the slope of the first line.
Step 2: Calculate the slope of the second line
Similarly, we calculate the slope of the second line using the points (8,12) and (x,24). We apply the same slope formula to these points. This will give us an expression for the slope m2 in terms of x.
Step 3: Apply the condition for perpendicular lines
Two lines are perpendicular if the product of their slopes is −1. We have calculated m1=31 and m2=x−812. We set their product equal to −1 to form an equation that we can solve for x.
Step 4: Solve for x
Now we solve the equation for x. We simplify the product of the slopes, then multiply both sides by (x−8), and finally isolate x to find its value. This gives us the required coordinate for the second line.