Line Problem Key

Problem:

Given line L

and points

which lie on line M.

If line M is perpendicular to line L, calculate the values of x and y.

A graphical investigation is shown to the below. Note the addition of point Q(q_x,q_y) which must lie on line L.

It appears we have a minimum of four (4) variables: y_A, x_B, q_x, and q_y.
So we'll need four equations.

Equation # 1 (relationship between x_B and y_A)

Given line L and points A and B:

We know the slope of line M must be perpendicular to L so...

so... an equation for line M is:

substituting in pont A(-10,y) which is on M yields:

substituting in point b(x, 4) which is on M yields:

solving for b in each equation and substituting yields:

relationship between y_A and x_B is:

Equation # 2 - The distances form point B (on line M) to line L must be twice the distance from point A (on line M) to line L. Lets use r as the ratio and set this ratio equal to 2.

so...

where:

and

substituting:

squaring both sides:

Equation # 3 (relationship of slopes for the points A, B, and Q... all on line M)

so...

Equation # 4 (specifies that point q must lie in line L)

Numerical Soution:

Using some initial guesses from our graphical solution:

Guess 1

checking answers:

Guess 2

checking answers:

Line L

Line M

Line M