Video Transcript
Given the following figure, find the numerical values of 𝑥 and 𝑦.
Well, to solve this problem, what we’re in fact going to do is use something called the side splitter theorem. And what this says is that if a line is parallel to a side of a triangle and the line intersects the other two sides, then this line divides those two sides proportionally. But what does this actually mean in practice?
Well, if we take a look at this example here, we’ve got our shape, which is shown. And then, we can see that the side 𝐴𝐶 divided by the side 𝐶𝐸 is gonna be equal to the side 𝐴𝐵 divided by the side 𝐵𝐷. Well, what we can do is in fact apply this to our problem. Because what we can say is that as the line 𝐸𝐷 is parallel to the line 𝐵𝐴, then 𝐶𝐸 over 𝐸𝐵 is gonna be equal to 𝐶𝐷 over 𝐷𝐴.
Well, for the left-hand side 𝐶𝐸 over 𝐸𝐵, we could write eight 𝑥 plus nine over five 𝑦 plus eight. However, so that we can solve to find 𝑥, what we want to do is just have the one variable. So what we can in fact write is eight 𝑥 plus nine over eight 𝑥 plus nine. And that’s because we’re told that 𝐶𝐸 is the same length as 𝐸𝐵. And then, for the right-hand side, we’re gonna have nine 𝑥 minus nine over eight 𝑥 plus three. Well, eight 𝑥 plus nine over eight 𝑥 plus nine is just gonna give us one. So we can rewrite our equation as one is equal to nine 𝑥 minus five over eight 𝑥 plus three.
Well, now, if we multiply through by eight 𝑥 plus three, what we’re gonna do is get eight 𝑥 plus three is equal to nine 𝑥 minus five. And then, if we subtract eight 𝑥 and add five to each side of our equation, then what we’re gonna get is eight equals 𝑥. And we can say that 𝑥 is equal to eight. So there we found 𝑥.
So now what we want to do is find 𝑦. And we can do that using a relationship we mentioned earlier. And that is that the length 𝐶𝐸 is the same length as the length 𝐸𝐵. So in fact, what we can do is equate eight 𝑥 plus nine and five 𝑦 plus eight. And what we can do now is substitute in our value for 𝑥, and that is that 𝑥 is equal to eight. And when we do that, we’re gonna have eight multiplied by eight plus nine equals five 𝑦 plus eight, which is gonna give us 73 is equal to five 𝑦 plus eight. So then, if we subtract eight from each side of the equation, we get 65 equals five 𝑦. Then, dividing through by five, we get 13 is equal to 𝑦. So therefore, we can say that the numerical values of 𝑥 and 𝑦 are 𝑥 equals eight and 𝑦 equals 13.
But what we could do now is a check, is actually use these numerical values we have for 𝑥 and 𝑦 to see if the side splitter theorem actually holds up with our shape. So I’m gonna clear a bit of space so that we can do this. Well, first, if we start with nine 𝑥 minus five, well, nine multiplied by eight is 72, minus five is 67. And then, if we move across to eight 𝑥 plus three or eight multiplied by eight is 64, add three is 67. Then, looking at eight 𝑥 plus nine, well, this is gonna be six more than 67, so it’s 73. And finally, five 𝑦 plus eight, well, we’ve got five multiplied by 13, which is 65, add eight is gonna give us 73.
Well, then, if we remember the side splitter theorem, we’re gonna have 𝐶𝐸 over 𝐸𝐵 is equal to 𝐶𝐷 over 𝐷𝐴. And that’s because we know that 𝐸𝐷 is parallel to 𝐵𝐴. So when we fill in the values that we’ve calculated, what we’re gonna have is 67 over 67 is gonna be equal to 73 over 73. Well, this is gonna give us one is equal to one, which in fact it is. So therefore, we can say that we found the correct numerical values of 𝑥 and 𝑦 because they uphold the side splitter theorem. And what’s interesting about this particular problem is that the lengths 𝐶𝐷 and 𝐷𝐴 are in fact the same and 𝐶𝐸 and 𝐸𝐵 are also the same.
