### Video Transcript

π, π, and π are three parallel planes intersected by two coplanar straight lines πΏ one and πΏ two such that π·π» over π»π equals one-third. If π΄πΆ equals 48 centimeters, find the length of line segment π΅πΆ.

Taking a look at our sketch, we see these three parallel planes π, π, and π. The straight lines πΏ one and πΏ two intersect these planes. And we see that the points of intersection create segments of these lines. On line one, thereβs a segment from point π· to point π», here, and then one from point π» to point π. On line two, thereβs a segment from π΄ to π΅ and then π΅ to πΆ. Regarding the relative lengths of the segments on line one, weβre told that the ratio of this line segment to this line segment is one-third. In other words, line segment π»π is three times as long as segment π·π».

Knowing this, and also knowing that line segment π΄πΆ is 48 centimeters, we want to find the length of line segment π΅πΆ. As weβve seen, thatβs this length here, along line two. The first thing we can realize is because these planes π, π, and π are parallel to one another, the ratio of line segment π·π» to π»π is the same as that ratio for the line segment π΄π΅ two π΅πΆ. This tells us that line segment π΅πΆ is three times as long as π΄π΅. More than this, itβs possible to write π΄π΅ in a different way. We can express it as the total line statement length π΄πΆ minus the distance of a line from point π΅ to πΆ.

So, we have this equation, and we want to use it to solve for the line segment π΅πΆ. To help us do that, letβs rearrange so that π΅πΆ is the subject. First, what weβll do is multiply both sides of the equation by π΅πΆ, and then weβll add this length to both sides so that negative π΅πΆ plus π΅πΆ on the left equals zero. On the right-hand side of this resulting equation then, we have four-thirds π΅πΆ so that if we multiply both sides by three-quarters, we have that line segment π΅πΆ equals three-quarters line segment π΄πΆ. And π΄πΆ, we recall, is 48 centimeters. If we substitute in this value, leaving out units, we find that π΅πΆ has a length of 36, and this length is in units of centimeters.

Our final answer then is that the length of the line segment π΅πΆ is 36 centimeters.