Determining slope of third line

Hi, Here's an interesting problem:

I have 2 lines with slopes and y-intercepts.

I have a third line with just a y-intercept and I need to determine it's slope in the following condition.

The condition:

What slope of the third line creates a triangle with an area of 0.1?

There should be two answers. One on each side of the intersection point of the first two lines.

How would I set this up to take in multiple line data? Can I reduce the calculation of the slope of the third line to one formula?

sample data:

Line 1: 2.41, 120.45

Line 2: 1.94, 134.26

Line 3: ?, 25.45

any help is appreciated...

Re: Determining slope of third line

You have three lines: y = 2.41x + 120.45, y = 1.94x + 134.26, and y = mx + 25.45

See where the three lines intersect, and that will give you three points. Find the distance between each point. Then use Heron's formula to calculate area.

Distance between two points :

Heron's formula for area of a triangle: where , and the three sides of the triangle have lengths .

Re: Determining slope of third line

Another way (without the distance formula and Heron's formula) is once you have the three points of intersection , the area is

Re: Determining slope of third line

Re: Determining slope of third line

still can't figure this out though, what's the answer?

Re: Determining slope of third line

Here are the three equations:

Let's figure out where the first two lines intersect. Then . We find . If we plug that value in for either line, we will get the y-coordinate. So, our first intersection point is .

The next point of intersection occurs when . That occurs at .

The last point of intersection occurs when . This occurs at .

So, those are your three points.

Now, plug those three points into the equation I gave you and solve for .

Re: Determining slope of third line

I need more than the slope for a triple intersection. I need the slope that produces a triangle of area 0.1. There should be 2 answers, One triangle on either side of the first intersection point of the two known lines.

Re: Determining slope of third line

The formula I gave you for area uses an absolute value. That means the two solutions are or .

Re: Determining slope of third line

I only have one intersection point though, too many unknowns

Re: Determining slope of third line

I gave you three intersection points. Read above. So, you have only one unknown: .