Video Transcript
Given that the line segment 𝐸𝐷 is parallel to the line segment 𝐶𝐵, find the value of 𝑥.
So, as we’ve been told in the question that the line segment 𝐸𝐷 is parallel to the line segment 𝐶𝐵, then what we’re gonna do is take a look at something called the side splitter theorem. And what the side splitter theorem tells us is that if a line is parallel to a side of a triangle and the line intersects the other two sides, then the line divides those sides proportionally.
But what does this mean in practice? Well, what we can do is apply it to the triangle that we have. So if we apply this to our triangle, we can say that 𝐴𝐸 over 𝐸𝐶 is equal to 𝐴𝐷 over 𝐷𝐵. And that is applying the side splitter theorem. We could even say that 𝐴𝐶 over 𝐴𝐸 is equal to 𝐴𝐵 over 𝐴𝐷 because this would also be highlighting the side splitter theorem. Okay, so now we have this, let’s use it to solve our problem and find the value of 𝑥.
As we’ve already stated, 𝐴𝐸 over 𝐸𝐶 is equal to 𝐴𝐷 over 𝐷𝐵. So therefore, what we can say is that log to the base three of 27 over log to the base three of three is equal to log to the base eight of 𝑥 over log to the base eight of eight. Well, in fact, what we could do now is rewrite this a little bit because we could say that log to the base three of 27 is log to the base three of three cubed. And then what we can do is in fact use one of our log rules. And that is that log to the base 𝑏 of 𝑀 to the power of 𝑘 is equal to 𝑘 log to the base 𝑏 of 𝑀.
Once we apply that, we can actually rewrite our equation. So what we’ll now have is three log to the base three of three over log to the base three of three is equal to log to the base eight of 𝑥 over log to the base eight of eight. Well, now there are a couple of ways we could carry out the next stage of the working.
Well, first of all, on the left-hand side, what we could do is just divide through by log to the base three of three, which would leave us with three multiplied by one over one. However, we could’ve also arrived at this by using one of our logarithm rules. And that is that log to the base 𝑏 of 𝑏 is equal to one. So therefore, log to the base three of three would be one. So it’ll be three multiplied by one over one.
Well, we can also see that on the right-hand side, we can also use that law because what we have is log to the base eight of eight as the denominator. So therefore, applying that law, we could say that this is going to be equal to one. So therefore, what we can do is rewrite our equation as three is equal to log to the base eight of 𝑥.
Well, now what we can do is actually convert three equals log to the base eight of 𝑥 into exponent form because we know that we have log to the base 𝑏 of 𝑀 is equal to 𝑎. Then 𝑏 to the power of 𝑎 is equal to 𝑀. Well, if we take a look at our equation, our 𝑎 would be three, our 𝑏 would be eight, and our 𝑀 would be 𝑥. So therefore, what we could say is that 𝑥 would be equal to eight cubed or eight to the power of three. So therefore, we can say that given that the line segment 𝐸𝐷 is parallel to the line segment 𝐶𝐵, then the value of 𝑥 is 512.
