Originally Posted by

**Grandad** Hello Aionios -

Two mathematical figures are said to be *similar* if they have exactly the same **shape**. (They're not necessarily the same size; if they are the same size as well, they are said to be **identical**, or **congruent**.)

So all *squares *will be similar, but *not* all *rectangles*. That's because all four sides of a square are the same length, but a rectangle can be long and thin, or short and fat.

You can always tell when you have two *similar triangles: *if their angles are the same as each other. For instance, if in triangle ABC, A = 90 deg, B = 60 deg and C = 30 deg, while in triangle PQR, Q = 90 deg, P = 60 deg and R = 30 deg, then the triangles are similar.

If you look at the diagram I attached with my first posting, you'll see some similar triangles. Triangle ABC is right-angled at A, and triangle ACD is right-angled at C. But that alone doesn't make them similar. So what about the other angles? Well, can you see why angle CAB = angle ADC? It's because when you add each one to angle DAC you get 90 degrees. Now if in two triangles, we have two pairs of equal angles, then the third pair must be equal, because all three angles in a triangle add up to 180 deg.

So, triangles ABC and DCA are similar. So what? Well, this means that their sides are *in the same ratio*. What do I mean by this? I'll come to that in a minute.

First, look at the original line, y = 12x/5 + 12. This has a gradient of 12/5, which means that as we move along the line, for every 5 units we move in the x-direction, we shall move 12 units in the y-direction. Now the line EAC is at right-angles to this line, so as we move along EAC, for every 12 units we move in the x-direction we shall move -5 units in the y-direction. So the *ratio* AB:BC = 12:5. By Pythagoras Theorem, this means that triangle ABC is a 5, 12, 13 triangle.

What this also means is that any other triangle that is *similar* to triangle ABC is also a 5, 12, 13 triangle. Aha! that means that triangle ACD is also a 5, 12, 13 triangle. So AD:AC = 13:5. But AC is 1 unit long, because we drew the lines DC and EF 1 unit away from y = 12x/5 +12.

So AD = 13/5 = 2.6 units. And so is AF, by the same reasoning.

Now use y = mx + c to find the equations of the lines DC and EF, and you're home and dry!

Hope that helps.

Grandad