 Consider the diagram of the triangles given on the screen here. Well, we have a couple triangles in play here, so we have the triangle A, B, C, which is a right triangle since angle C is a right angle. And we know that the measure of angle A is 38 degrees. But we also have the right triangle C, D, B right here, where the angle C, of course, is still right, it's shared. And we get the measure of angle D is 45 degrees in that situation. We know that the distance between points A and D is 32 units. And we're asked to figure out what is the distance between C and D? That is, what's the length of this line segment right here? So since that's the value we want to find, let's give it a name, let's call it X. And so how can we relate together the measurements of the segment AD with the measures of 38 degrees and 54 degrees? What can we do there? Well, in terms of the line segment AC, we know that's going to equal X plus 32 because we know that distance here. So AC, which is the side length of a right triangle, right? It's related to this triangle here. But look at the ABC triangle, we have X plus 32, we have 38 degrees for angle A. We also know that X is the side, the adjacent side of the right triangle, BCD with respect to angle D right there. And so we could try to use some type of right triangle trigonometry to relate those things together. Notice that DC is equal to X. But the problem is we don't know this side right here, right? This, this, the side BC, which is connected to both of those triangles. So it's called Y for a moment. That's a shared common side. And so maybe we can utilize that to try to solve for this variable X. So if you look at the little triangle, for example, the triangle DCB. And we think of a right triangle ratio there, we could use tangent. Notice if we use opposite over adjacent, this is going to equal tangent of 54 degrees. Right, opposite over adjacent like so. Also, if we look at the larger triangle, the ABC triangle, if we also did a tangent ratio there, we're going to get Y over X plus 32 is equal to tangent of 38 degrees. You'll notice with both of these equations, if you were to clear the denominators, you would end up with the statement that X times tangent of 54 degrees. That's equal to Y. But Y is also equal to X plus 32 times tangent of 38 degrees. Basically, if you cleared the denominators for the second one, you get this right here. And so Y here is sort of acting like a middleman. If we just take it out, we now have an equation of X that we can try to solve for. So to proceed forward, we're going to distribute the tangent of 38 degrees. We're going to postpone the approximation of tangents at this moment. We'll wait until later on to avoid rounding errors. We're going to get X times tangent of 54 degrees. Always remember to write the degree symbol here. You're going to get X times tangent of 38 degrees. And then we're going to get plus 32 tangent of 38 degrees. So we want to combine the tangents together. So I'm going to subtract X times, excuse me, we don't want to combine the tangents together. We want to combine X together. So subtract X times tangent of 38 degrees from both sides of the equation. This will then give you on the right hand side X times tangent of 54 degrees minus X times tangent of 38 degrees. And this is going to equal 32 times tangent of 38 degrees on the right hand side. Factor the left hand side, that is, there's a common factor of X. So we're going to get tangent of 54 degrees minus tangent of 38 degrees. This is still equal to 32 times tangent of 38 degrees. And so then if we divide by the coefficient of X, which in this case is tangent of 45 degrees minus tangent of 38 degrees, we get the exact answer for X. We get that X is going to equal 32 times tangent of 38 degrees over tangent of 54 degrees minus tangent of 38 degrees, like so. And so if you solve this equation a little bit differently, your final answer might have looked a little bit different, but it would be equivalent to the answer that you now see here on the screen. So we have the exact answer. Now, probably we need an approximation of this. And so you'll notice that we have done no estimates so far. We have the exact measures of tangent of 38 degrees and tangent of 54 degrees. That's one reason we haven't done anything with it yet. So now we're going to throw all of this in our calculator. And this is going to be a chore of calculator syntax. And so I can't exactly tell you how to do it with your calculator, because the syntax of different calculators vary by brand. So consult your user manual in that regard. But if you put this into your calculator correctly, we should see how many places of decimal approximation do we need? Do we need like one decimal place, accurate two decimal place? Are we working with significant digits? Again, follow the instructions you would have in that situation. But ignoring all of those at this point, we're going to round to the nearest whole number. And we see in fact that this number turns out to be 42. So the distance between point C and D in the original diagram is going to be approximately 42 units.