This lesson will cover a few simple examples related to what I covered recently.
Example 1 Show that the points (1, 2) and (3, 4) lie on the opposite sides of the line x + y – 4 = 0.
Solution Easy! All you have to do is put the values of the coordinates in the given equation and check the signs.
For the first point 1 + 2 – 4 = – 1 < 0, and for the second, 3 + 4 – 4 = 3 > 0. Opposite signs !
Therefore the two points lie on the opposite sides of the line. Here’s how things look..
Example 2 Show that origin lies inside the triangle whose sides are x + y = 4, y – 2x = 4 and x – 4y = 4.
Solution Hmm.. this one requires a little more thinking. I’ll show you the figure first.
Now here’s the trick. Observe that the origin (or any other point inside the triangle) has this property: It will be on the same side with respect to any given side, as the vertex opposite to that side.
In other words, O and A will be on the same side of BC, O and B will lie on the same side of CA, and O and C will lie on the same side of AB.
If either of these three conditions is not met, then the point O, will lie outside the triangle. Try and figure this out yourself, should be easy.
Now all we have to do is check for these three conditions. I’ll make a small table to show this. Note that we’ve to change the equations to the form ax + by + c = 0 before checking the signs.
|Side||Sign w.r.t Opposite Vertex||Sign w.r.t O|
|AB: x + y – 4 = 0||(-60/7) -ve||(-4) -ve|
|BC: x – 4y – 4 = 0||(-20) -ve||(-4) -ve|
|CA: -2x + y – 4 = 0||(-12) -ve||(-4) -ve|
And that’s it, we got what we needed !
O and A lie on the same side of BC, O and B lie on the same side of CA, and O and C lie on the same side of AB.
Therefore the point O lies inside the triangle.
That’s about enough for this topic. I’ll talk about family of lines in the next lesson. Meet you there !