# Question Video: Finding an Unknown Length of a Straight Line Using the Properties of the Parallel Planes and the Coplanar Straight Lines Mathematics

π, π, and π are three parallel planes intersected by two coplanar straight lines πΏβ and πΏβ such that π·π»/π»π = 1/3. If π΄πΆ = 48 cm, find the length of line segment π΅πΆ.

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### Video Transcript

π, π, and π are three parallel planes intersected by two coplanar straight lines πΏ one and πΏ two such that π·π» over π»π equals one-third. If π΄πΆ equals 48 centimeters, find the length of line segment π΅πΆ.

Taking a look at our sketch, we see these three parallel planes π, π, and π. The straight lines πΏ one and πΏ two intersect these planes. And we see that the points of intersection create segments of these lines. On line one, thereβs a segment from point π· to point π», here, and then one from point π» to point π. On line two, thereβs a segment from π΄ to π΅ and then π΅ to πΆ. Regarding the relative lengths of the segments on line one, weβre 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΄πΆ is 48 centimeters, we want to find the length of line segment π΅πΆ. As weβve seen, thatβs 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΅πΆ. This tells us that line segment π΅πΆ is three times as long as π΄π΅. More than this, itβs possible to write π΄π΅ in a different way. We can express it as the total line statement length π΄πΆ minus the distance of a line from point π΅ to πΆ.

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