Video Transcript
Find the values of ๐ฅ and ๐ฆ.
What we see in the figure is the larger triangle ๐ด๐๐ท, where the line segments ๐ธ๐ต and ๐น๐ถ are cutting this triangle. In addition to that, the line segments ๐น๐ถ, ๐ธ๐ต, and ๐ด๐ท are all parallel. And by the side splitter theorem, when a triangle is cut by a parallel line to one of the sides, it splits the side lengths proportionally. We see that the three segments created, ๐ธ๐ท, ๐น๐ธ, and ๐๐น, are all equal to each other. And since thatโs the case, and since these parallel lines cut this triangle proportionally, we can say that ๐๐ถ is going to be equal to ๐ถ๐ต, which is going to be equal to ๐ต๐ด.
If we write the statement like this, ๐ฆ plus four equals four ๐ฅ plus one which equals ๐ฅ squared minus four, it doesnโt seem really clear what weโre solving for. So letโs break this up. If we said four ๐ฅ plus one equal to ๐ฅ squared minus four, we can solve for ๐ฅ.
Since we have ๐ฅ squared, weโre dealing with a quadratic equation. And we want to set this equation equal to zero. So we subtract four ๐ฅ from both sides of the equation. And then we subtract one from both sides of the equation. On the left, weโll only have zero. And on the right, weโll have ๐ฅ squared minus four ๐ฅ. And then weโll have minus four minus one, which we can combine to minus five, so that we have zero equals ๐ฅ squared minus four ๐ฅ minus five.
And then we can factor this quadratic. We need to find two terms that multiply together to equal negative five and add together to equal negative four, which will be positive one and negative five. Then we set both of these terms equal to zero so that we have ๐ฅ plus one equals zero and ๐ฅ minus five equals zero. Weโll either have ๐ฅ equals negative one or ๐ฅ equals positive five.
If we plug in ๐ฅ equals negative one into four ๐ฅ plus one and ๐ฅ squared minus four, we get negative lengths, which wouldnโt work. And that means the only valid option for us is ๐ฅ equals five. If ๐ฅ equals five, then these set segments are equal to 21. Four times five plus one equals 21, and five squared minus four equals 21.
Now that we know that, we can create a third equation to solve for ๐ฆ. This is because ๐ฆ plus four must also be equal to 21. If we write this, 21 equals ๐ฆ plus four, and then subtract four from both sides, we see that ๐ฆ must be equal to 17. Because we knew that these parallel lines cut this triangle proportionally, we were able to find that ๐ฅ equals five and ๐ฆ equals 17.
