Question

A line segment is bisected by a line with the equation `8 y + 5 x = 4`. If one end of the line segment is at `( 1 , 3 )`, where is the other end?

Answer 1

Other end coordinates `(-1.809, -1.4944)`

Explanation:

Assumption: It’s a perpendicular bisector that intersects the line segment.
Slope of perpendicular bisector m1;
`y=-(5/8)x+(1/2)`
`m1=-(5/8)`

Slope of line segment `m2=-(1/m1)=-(1/-(5/8))=8/5`
Equation of line segment :
`y-3=(8/5)(x-1)`
`5y-15=8x-8`
`5y-8x=7color(white)(aaaaa)Eqn (1)`
`8y+5x=4color(white)(aaaaa)Eqn (2)`

Solving Eqns (1) & (2) to get the midpoint coordinates,
`25y-40x=35`
`64y+40x=32` color(white)(aaa) Adding both,
`89y=67 or y=67/89=0.7528`
`x=(4-(8*(67/89)))/5=-0.4045`
Midpoint coordinates (-0.4045,0.7528)

`—0.4045=(x1+1)/2`, `0.7528=(y1+3)/2`
`x1=-1.809, color(white)(aaaaa)y1=-1.4944`