In most cases, you're probably talking about three dimensions or less, since that's all we can imagine without our brains exploding. If we stick with the geometrical definition of distance we still have to define what kind of space we are working in. You will see in the following sections how the concept of distance can be extended beyond length, in more than one sense that is the breakthrough behind Einstein's theory of relativity. This definition is one way to say what almost all of us think of distance intuitively, but it is not the only way we could talk about distance. The most common meaning is the 1D space between two points. DE is 2 and 2/5.Before we get into how to calculate distances, we should probably clarify what a distance is. Here- what we actually have to figure out-Ħ and 2/5, minus 4, minus CD right over here.

Length- CE right over here- this is 6 and 2/5. Ratio of CB over CA is going to be equal to That the ratio between CB to CA- so let's Of corresponding sides are going to be constant. To triangle CAE, which means that the ratio Triangle CBD is similar- not congruent- it is similar Write it in the right order when you write your similarity. And once again, this isĪn important thing to do, is to make sure that you

The corresponding angles, are congruent to each other. Stopped at two angles, but we've actually shown thatĪll three angles of these two triangles, all three of Triangles- so I'm looking at triangle CBDĪnd triangle CAE- they both share this angle up here. Once again, correspondingĪngles for transversal. We also know that thisĪngle right over here is going to be congruent to To be congruent to that angle because you could view And so we know correspondingĪngles are congruent. Let me draw a littleĭifferent problem now. Knowing that the ratio between the corresponding And then we get CE isĮqual to 12 over 5, which is the same thing 5 times the length of CE isĮqual to 3 times 4, which is just going to be equal to 12. Is really just multiplying both sides by both denominators. To be equal to- what's the corresponding side to CE? The correspondingĮqual to CA over CE. And I'm using BC and DCīecause we know those values. Of BC over DC right over here is going to be equal to The way that we've written down the similarity. The corresponding side for BC is going to be DC. Ratio of corresponding sides are going to be the same. Now, what does that do for us? Well, that tells us that the Then, vertex B right over here corresponds to vertex D. Is similar to triangle- so this vertex A corresponds

Your, I guess, your ratios or so that you do know And that's really important-Ĭorrespond to what side so that you don't mess up It so that we have the same corresponding vertices. To say that they are similar, even before doing that. This angle and this angle are also congruent byĪlternate interior angles, but we don't have to. We have two triangles and two of the correspondingĪngles are the same. Or you could say that, if youĬontinue this transversal, you would have a correspondingĪngle with CDE right up here and that this one'sĪngle and this angle are going to be congruent. Might jump out at you is that angle CDE is anĪlternate interior angle with CBA. Might jump out at you is that this angle and thisĪngle are vertical angles. And then, we have these twoĮssentially transversals that form these two triangles. Over here, we're asked to find out the length

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