Question

A bridge being designed will cross the river at a right angle. The equation of the left bank of the river is j = 2x+8. The center of the bridge will pass through (0,2). What is the equation of the line representing the bridge?​

Answer 1

Answer:   y = (-1/2)x + 2

This is the same as y = -0.5x+2

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Explanation:

Refer to the diagram below.

The blue line is the graph of y = 2x+8, which is the left bank of the river.

For now, ignore the y = (-1/2)x+2 in the diagram and pretend we don't know that just yet. The red line is the bridge and it's some linear equation in the form y = mx+b. We're told the bridge is perpendicular to the river, so the bridge must also be perpendicular to the riverbank.

The blue line has slope m = 2. Apply the negative reciprocal to this to end up with -1/2. This is the perpendicular slope, and the slope of the red line. Note how the two slopes (2 and -1/2) multiply to -1. Any two perpendicular lines will have their slopes multiply to -1, as long as neither line is vertical or horizontal.

Since we want the bridge to pass through (0,2), this must mean the y intercept is b = 2.

So with m = -1/2 and b = 2, we go from y = mx+b to y = (-1/2)x + 2 which is the equation of the bridge shown in red.

This is the same as y = -0.5x + 2 since -1/2 = -0.5