𝑋, 𝑌, and 𝑍 are three parallel planes intersected by two coplanar straight lines 𝐿 one and 𝐿 two such that 𝐷𝐻 over 𝐻𝑂 equals one-third. If 𝐴𝐶 equals 48 centimeters, find the length of line segment 𝐵𝐶.
Taking a look at our sketch, we see these three parallel planes 𝑋, 𝑌, and 𝑍. The straight lines 𝐿 one and 𝐿 two intersect these planes. And we see that the points of intersection create segments of these lines. On line one, there’s a segment from point 𝐷 to point 𝐻, here, and then one from point 𝐻 to point 𝑂. On line two, there’s a segment from 𝐴 to 𝐵 and then 𝐵 to 𝐶. Regarding the relative lengths of the segments on line one, we’re told that the ratio of this line segment to this line segment is one-third. In other words, line segment 𝐻𝑂 is three times as long as segment 𝐷𝐻.
Knowing this, and also knowing that line segment 𝐴𝐶 is 48 centimeters, we want to find the length of line segment 𝐵𝐶. As we’ve seen, that’s this length here, along line two. The first thing we can realize is because these planes 𝑋, 𝑌, and 𝑍 are parallel to one another, the ratio of line segment 𝐷𝐻 to 𝐻𝑂 is the same as that ratio for the line segment 𝐴𝐵 two 𝐵𝐶. This tells us that line segment 𝐵𝐶 is three times as long as 𝐴𝐵. More than this, it’s possible to write 𝐴𝐵 in a different way. We can express it as the total line statement length 𝐴𝐶 minus the distance of a line from point 𝐵 to 𝐶.
So, we have this equation, and we want to use it to solve for the line segment 𝐵𝐶. To help us do that, let’s rearrange so that 𝐵𝐶 is the subject. First, what we’ll do is multiply both sides of the equation by 𝐵𝐶, and then we’ll add this length to both sides so that negative 𝐵𝐶 plus 𝐵𝐶 on the left equals zero. On the right-hand side of this resulting equation then, we have four-thirds 𝐵𝐶 so that if we multiply both sides by three-quarters, we have that line segment 𝐵𝐶 equals three-quarters line segment 𝐴𝐶. And 𝐴𝐶, we recall, is 48 centimeters. If we substitute in this value, leaving out units, we find that 𝐵𝐶 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 𝐵𝐶 is 36 centimeters.