Grades 11-12 Video Solutions 2022-2022_11-12

Video Transcription

Two circles cut a rectangle AFMG as shown. The line segments outside the circles have lengths AB equals 8, CD equals 26, EF equals 22, GH equals 12, and JK equals 24. What is  the length of LM? Here's the diagram a little bit bigger. So let's notice a curious thing about these circles. They cut out a few more segments  that we don't really know the lengths of. But what we do know is that these segments are aligned with each other, meaning if we draw a vertical line through the centers of  the circles, it's going to go through their midpoints of these unknown segments. So now let's just give some segments some variable names. I'm going to call this one A, this  is B and B, because this segment is cut in half by this line, C, DD, E, F, GG, H, I, I, and J. So this also cuts up our big rectangle into three smaller rectangles, here, here,  and here. Now since they're rectangles, we know that their opposite sides are equal, so A plus B has to be equal to F plus G. Similarly, B plus C plus D has to equal G  plus H plus I, and then D plus E is equal to I plus J, so I've just written that down over here. So now I want to add up these three equations, but I want to do it in such a way  that we can eventually eliminate all these unknown lengths, because we don't know what B, D, G, and I are. So here's an A plus B, and here's a B plus C plus D, and I want them  to be on opposite sides of the equation, so I'm going to add up these two as they're written,  and then here's a D on this side and a D on this side. This is on the left side and this is on the right side, so I'm going to add this one in as it's written as well to get  this big mess. But the nice thing is that there's one B here, one B here, a G here, a G here, I, I, D, D. And so now we can cancel them out to get that A plus H plus E equals  F plus C plus J. Now we know what five of these quantities are, the only unknown one is J, so now we can just subtract over the F and the C to figure out what J is. So A  plus H plus E is equal to 8 plus 24 plus 22, which is 54, and then minus the F and the C gets us 54 minus 26 minus 12, which is 16, so this length has to be equal to 16.

Video Summary

The problem involves determining the length of a segment within a rectangular shape cut by two circles, with given segment lengths. By assigning variables to the additional unknown segments, equations are formed using the properties of rectangles and circles. Through carefully adding and manipulating these equations, we eliminate unknowns and solve for the necessary segment. The calculation shows that the length of LM is 16, after balancing known side lengths and subtracting specific given values.