Say I have a large mirror and in front of the mirror, there is an object and I'm standing at a different position in front the mirror. At what position should I look in the mirror so that I can see the object in the mirror?
For example: If I stand at the point $(2,6)$ and object is at $(4,9)$ and the mirror is on the x-axis and has infinite length. What position should I look at to see the object?
The basic idea is, you can think of the mirror as a window, when you look through it, you see the symmetric version of your own dimension. If the mirror is on the x-axis, the objects distance to the mirror will be the same on both sides;
if the object is at (4,9), its distance to the mirror is 9 units, meaning it will be 9 units away from the 'window' when you look out. So the objects 'virtual position' will be at (4,-9), that's where you will see it. Similarly you will see yourself at (2, -6).
Notice that this means while the actual object is on the left of your real position, the virtual object is on the right of your virtual position.
Where you need to look on the mirror is now a geometry problem. Draw this system with both the real and virtual points on the coordinate system, then draw a line straight from your real position to the virtual position of the object. Then draw a right triangle with this line being the hypotenuse, and the other side of it connecting to your real position perpendicularly.
What we need to find is where on the x-axis the hypotenuse you drew is passing from, that's the point where you'll look to see the object. To find this point we use the following ratio;
$$ \frac{2}{15} = \frac{d}{6} $$ This can be written since the angle of the triangle is constant, even if we slice the triangle with the x-axis, this means the change of the length of the short side of the triangle with respect to the long side, is a constant ratio. So if it changed from 0 to 2 when the long side changed from 0 to 15, it will have changed with the same ratio from 0 to d when the long side changed from 0 to 6. So we get
$$ d = \frac{12}{15} = 0.8 $$
So the point where the hypotenuse intersects the x-axis is (2.8, 0), that is where the mirror 'cuts' your line of sight straight to the object, so that's where you have to look.
