A line segment is bisected by a line with the equation 4 y + 9 x = 8 . If one end of the line segment is at (5 ,2 ), where is the other end?

1 Answer
Douglas K.
Oct 19, 2016

The other end is at (1295/97, 554/97 )

Explanation:

Write the given line in slope-intercept form:

y = -9/4x + 2

Because bisector is perpendicular, the slope of the line segment will be the negative reciprocal of its bisector, 4/9.

[This reference gives us an equation for the distance from the point to the line.](https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line) The distance from the point to the line is:

d = |(4(2) + 9(5) - 8)/(sqrt(4^2 + 9^2))| = 45sqrt(97)/97

The length of the line segment is twice this distance, 90sqrt(97)/97.

From point (5,2), we move to the right a distance, (x), and up a distance y , we know that y is 4/9x, and we know that length of the hypotenuse formed by this right triangle is 90sqrt(97)/97

(90sqrt(97)/97)^2 = x^2 + (4/9x)^2

90^2/97 = 81/81x^+ 16/81x^2

x^2 = 81(90^2)/97^2

x = 810/97

y = 360/97

The other end of the line is at point:

(5 + 810/97, 2 + 360/97) = (1295/97, 554/97 )

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