 Suppose we know the diagonals of a parallelogram are given as 24.2 centimeters and 35.4 centimeters. Just as a reminder, a parallelogram is going to be a four-sided polygon aka a quadrilateral where opposite sides are parallel. So these two lines are parallel lines and then these two sides of the parallelogram are parallel as well. So the diagonals are going to be the distances from opposite corners of our quadrilateral right here. Suppose one of them is 24.2 which means if you cut that in half this side would be 12.1 and then this side right here this diagonal is 35.4 centimeters which if you cut in half you'll get 17.7 centimeters for that length right there the half diagonal and suppose that the angle between the two diagonals is 65.5 degrees as illustrated here in the diagram. With this information in mind can we figure out the dimensions of the parallelogram can we find the sides of the parallelogram and let's actually start with this side right here which we'll call it dimension x. Notice that we have a side angle side situation here so we can use the law of cosines to figure out the missing side x right here. So we get that x squared is going to equal 17.7 squared plus 12.1 squared minus 2 times 17.7 times 12.1 times cosine of the angle 65.5 degrees. So there's some huge number crunching to do on the other side but that's what calculators are for right. 17.7 squared gives us 313.29 12.1 squared gives us 146.41. We then take 2 times 17.7 times 12.1 that gives us 428.34 and then we have to compute cosine of 65.5 degrees I'll do that one in just a second. So let's take the 12.1 squared plus the 17.7 squared that adds up to give us approximately 459.7 and then we plug in cosine of 65.5 into our calculator times that by the 428.34 we're going to get negative 177.63 which then gives us 282.07 again somewhat of another crunch here that gives us x to give us excuse me that gives us x square to get x we need to take the square root so x is going to be approximately the square root of 282.07 which is going to give us 16.8 centimeters that's the side length on that side right there so 16.8 to do the other one uh well we just have to figure out this angle right here this angle is going to be supplementary to the other angle so we could do 180 degrees minus 65.5 we could do that turns out we don't even need to know what it is because if you take cosine of 180 degrees minus theta that is if you take the supplement that's just equal to negative cosine of theta so we can actually get away with the 65.5 we're just going to switch the signs so to find y squared we just have to take the 17.7 squared plus the 12.1 squared then we're going to get plus 2.17.7 times 12.1 times cosine of 65.5 degrees and so that's going to simplify dramatically in the end it's basically just going to give you this 459.7 plus uh 17.63 so what I'm trying to say is the calculation can be basically the same except that difference becomes a plus now and so in this situation you end up with 637.33 and then to find y we have to take the square root of that y will be the square root of 637.33 which is approximately 25.2 centimeters like so so using the law of cosines we can find the dimensions of a parallelogram if we know the lengths of the of the diagonals and we know the angle between the two