Grades 9-10 Video Solutions 2014-Grades 9-10 Video Solutions 2014 problem6

Video Transcription

Question number six. The length of the edge of the big regular hexagon is two times the length of the edge of the small regular hexagon. The small regular  hexagon has an area of four centimeters squared. What is the area of the big hexagon? There are two ways to solve this problem.  One of them requires that we know how to compute the area of the hexagon of a regular hexagon.  And so let me just recall that formula. If we know that the edge here has length s,  then the area is given by three root three over two times s squared. And so we see that  if the short edge here on the smaller hexagon has, for example,  uh, length is equal to l, if s is equal to 2l, then the area  is four times as much.  Okay, so we have three root three over two, and now 2l quantity squared gives us three root three  over two times l squared and then times two squared, so times four. So this is four times  the area of the small hexagon.  And we can choose our answer now. Four times the original area is four times  four centimeters squared, or that is a 16 centimeters squared. Another way to solve the problem, which is a lot shorter but also slightly more difficult, is that in two dimensions,  uh, changing one dimension by a factor of two,  so increasing a dimension by a factor increases the area  by the square of that factor  so this works for triangles rectangles any shape that is two-dimensional and general will have some sort of a square here in the formula for its area and so  doubling that dimension will cause a factor of two to appear twice so we will have four times as much area still the answer is 16 centimeters squared

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