Question

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?

Answer 1

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. 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 )`