Question

When the point ( 5 , 5 ) is reflected about a line l, the result is point ( − 1 , 11 ) . What is the slope-intercept form of the equation for line l?

Answer 1

The slope-intercept form of the equation for line l that reflects the point (5, 5) to (-1, 11) is y = x + 6.

Step-by-step explanation:

To find the slope-intercept form of the equation for line l that reflects the point (5, 5) to (-1, 11), we can first determine the midpoint between these two points, which lies on line l. The midpoint M is given by the average of the x-coordinates and the y-coordinates of the two points. This yields M = ((5 + (-1)) / 2, (5 + 11) / 2) = (2, 8).

The slope of line l is perpendicular to the slope of the line connecting the two points because reflection over a line implies that line is the perpendicular bisector of the segment joining the original and reflected points. The slope between (5, 5) and (-1, 11) is (11 - 5) / (-1 - 5) = 6 / -6 = -1. A line perpendicular to this has a slope that is the negative reciprocal, so the slope of line l is 1.

The equation of a line with slope m and passing through a point (x1, y1) is given by y - y1 = m(x - x1). Applying the point-slope form with our midpoint (2, 8) and slope 1 gives us the equation of line l as y - 8 = 1(x - 2).

Simplifying this into slope-intercept form yields y = x + 6. Therefore, the slope-intercept form of the equation for line l is y = x + 6.

Answer 2

The slope-intercept form of the equation for line l is y = -x + 10.

Step-by-step explanation:

The slope-intercept form of an equation for a line is y = mx + b, where m represents the slope of the line and b represents the y-intercept. To find the equation of the line reflecting the point (5, 5) about line l, we can use the formula:

m = (y2 - y1) / (x2 - x1)

Plugging in the given points, we have:

m = (11 - 5) / (-1 - 5) = 6 / -6 = -1

Since the line is reflected, the slope remains the same, but the y-intercept changes. We can use the point-slope form of a line to find the new equation:

y - y1 = m(x - x1)

Substituting the given point (5, 5), we have:

y - 5 = -1(x - 5)

Simplifying, we get:

y - 5 = -x + 5

Adding x to both sides and adding 5 to both sides, we get:

y = -x + 10

Therefore, the slope-intercept form of the equation for line l is y = -x + 10.