### Video Transcript

Find the values of π₯ and π¦.

What we see in the figure is the larger triangle π΄ππ·, where the line segments πΈπ΅ and πΉπΆ are cutting this triangle. In addition to that, the line segments πΉπΆ, πΈπ΅, and π΄π· 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, πΈπ·, πΉπΈ, and ππΉ, are all equal to each other. And since thatβs the case, and since these parallel lines cut this triangle proportionally, we can say that ππΆ is going to be equal to πΆπ΅, which is going to be equal to π΅π΄.

If we write the statement like this, π¦ plus four equals four π₯ plus one which equals π₯ squared minus four, it doesnβt seem really clear what weβre solving for. So letβs break this up. If we said four π₯ plus one equal to π₯ squared minus four, we can solve for π₯.

Since we have π₯ squared, weβre dealing with a quadratic equation. And we want to set this equation equal to zero. So we subtract four π₯ from both sides of the equation. And then we subtract one from both sides of the equation. On the left, weβll only have zero. And on the right, weβll have π₯ squared minus four π₯. And then weβll have minus four minus one, which we can combine to minus five, so that we have zero equals π₯ squared minus four π₯ 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 π₯ plus one equals zero and π₯ minus five equals zero. Weβll either have π₯ equals negative one or π₯ equals positive five.

If we plug in π₯ equals negative one into four π₯ plus one and π₯ squared minus four, we get negative lengths, which wouldnβt work. And that means the only valid option for us is π₯ equals five. If π₯ 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 π¦. This is because π¦ plus four must also be equal to 21. If we write this, 21 equals π¦ plus four, and then subtract four from both sides, we see that π¦ must be equal to 17. Because we knew that these parallel lines cut this triangle proportionally, we were able to find that π₯ equals five and π¦ equals 17.