I used to be quite good at Maths in school. There were some things however which I never quite got. Similar triangles were one of them. From what I remember, if you knew one triangle, you could employ that to calculate another triangles dimensions.
So, how do similar triangles work ( wiki just gave me a headache)? Any real world uses.
If two triangles are similar, then the ratios between corresponding sides are the same. You can use this to calculate the length or height of something indirectly.
Take a tall building on level ground and imagine a triangle composed of the top and bottom corner of the building the furthest from the sun and the shade of the top corner. This triangle will be similar to one made by the same rules with a stick. Measure the length of the shades on the ground, call them sb for the shade of the building and ss for the shade of the stick. Measure the height of the stick, call it hs. Then height of the building can be found by comparing the ratios, which have to follow hb/hs = sb/ss -> hb =hs * sb/ss
Here’s a good website on the subject with a lot of pictures. (Pictures are primary here; words are used to point out things in the pictures you haven’t noticed yet. The Pythagorean Theorem was proven using a picture and a word, and the word was superfluous.)
In that spirit, here’s an illustrated example of using similar triangles to measure how tall a building is. You’ll note all of this reduces to algebra, but that’s how you get machines to solve the numerical parts of these problems for you; understanding is done using the images.
Consider two triangles T, T’ with angles θ[SUB]1[/SUB], θ[SUB]2[/SUB], θ[SUB]3[/SUB] and θ’[SUB]1[/SUB], θ’[SUB]2[/SUB], θ’[SUB]3[/SUB]. Similarity means that T’ can be obtained from T by scaling, rotation, reflection, and translation. The latter three operations don’t affect the lengths of sides, and the three angles of the triangles are unaffected by all four operations. It follows that the triangles T, T’ are similar iff (after permuting the angles appropriately) we have θ[SUB]1[/SUB] = θ’[SUB]1[/SUB], θ[SUB]2[/SUB] = θ’[SUB]2[/SUB], and θ[SUB]3[/SUB] = θ’[SUB]3[/SUB]. The corresponding sides are also proportional, since only scaling affects the side lengths. Thus if x[SUB]i[/SUB] and x’[SUB]i[/SUB] denote the sides opposite θ[SUB]i[/SUB] and θ’[SUB]i[/SUB], respectively, then x’[SUB]i[/SUB]/x[SUB]i[/SUB] = t is independent of i; it’s the factor by which T is scaled to produce T’. Furthemore, the area of T’ is t[SUP]2[/SUP] times the area of T, since scaling by t changes area by a factor of t[SUP]2[/SUP].
Perhaps also worth mentioning, a triangle can be defined with a range of things. Three lengths gives you a triangle, as does three points (really the same thing, but the points locate it as well), two angles and a length, two lengths and an angle. Two angles define the shape, but not the size. The trick with similar triangles is to find a way of showing that two triangles are similar. So if they have a shared angle, you are onto a good thing. You might be able to construct other relationships that show another pair of angles must be the same - at which point you know the triangles are similar. Then you can go on to find the ratio of sizes and thus construct the entire second triangle in terms of the first. This is thus a powerful technique in constructing a solution. It may be the entire solution, or a tool used on the way to solving a more messy problem.
Two lengths and a NON-shared angle won’t necessarily give you similar triangles (the dreaded SSA situation)…
Now, I get to what my teenage self could never comprehend, what exactly makes triangles similar?
Two triangles are similar if they have the same angles (which is what Itself said in a bit more detail).
If two triangles have two angles that are the same (and as two angles determine the third, hence three angles that are the same), they are similar.
Informally: they’re the same shape, just not the same size. So all the lengths are twice as long (or 5 times as long, or 15% as long, or whatever) in one triangle as in the other. If you put a triangle into a copy machine and set it to enlarge or reduce, or you zoom out or zoom in, you get a similar triangle; but if you stretch it in one direction but not another, or look at it in a funhouse mirror, it’s not similar.
I think most people have an intuitive idea of similar triangles, even if they don’t realize it.
Imagine you had a triangle, and while you were sleeping, God replaced the Universe with a replica where the linear size of everything was doubled. When you woke up, you wouldn’t have anyway to tell.
In particular, you wouldn’t be able to tell your new doubled triangle from the old one. A protractor would give you the same angles. You could use a ruler, but since all the rulers have doubled in size, you couldn’t tell that way either. All the sides would be the same size relative to the ruler, and relative to each other.
That’s the basic idea of similar shapes. There’s no natural “independent” length in geometry. Unless you have a way to compare them to each other or a third thing, a big version of a shape is the same as a small version.