tag:blogger.com,1999:blog-5541500.post503335932114356886..comments2021-06-15T22:09:34.885+09:00Comments on BigHominid's Hairy Chasms: this week's MGRE Math Beast ChallengeKevin Kimhttp://www.blogger.com/profile/01328790917314282058noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-5541500.post-35863565197220043412011-12-06T11:59:48.619+09:002011-12-06T11:59:48.619+09:00Based on the figure, a right triangle, we&#39;re c...Based on the figure, a right triangle, we&#39;re comparing the quantities abc and h(a^2 + b^2). Right away, thanks to Uncle Pythagoras, we know that the parenthetical expression can be converted to c^2.<br /><br />Let&#39;s set up a &quot;mysterious&quot; inequality in which the letter Q represents &quot;greater than, less than, or equal to.&quot; The first step of this mysterious inequality would be:<br /><br />abc Q h(c^2) [based on our deductions]<br /><br />We can then divide by c on both sides, giving us<br /><br />ab Q ch<br /><br />The length h is perforce less than either a or b. The length c, meanwhile, is perforce greater than a or b, but not necessarily greater than ab.<br /><br />Hm.<br /><br />Without actually looking up the properties of right triangles, I&#39;d have trouble seeing how this problem could be solved without more information. So unless someone can swoop in with a better notion, I&#39;m leaning toward (D): the relationship can&#39;t be determined from the information given.<br /><br />But let&#39;s try something. Let&#39;s divide c into subsections x and y. Let&#39;s further assume a and b are equal, thus making x and y equal to each other. Length c can, in such a situation, be rewritten as 2x (or 2y). In this situation, h is at its maximal length relative to the legs and the hypotenuse, and we have a 45-45-90 triangle.<br /><br />C can be rewritten as a√2.<br />X and Y can each be rewritten as (a√2)/2.<br />H can also be rewritten as (a√2)/2, because a 45-45-90 triangle is <i>half of a square.</i> The altitude becomes half a diagonal, just as X and Y are halves of a diagonal.<br /><br />So:<br /><br />abc = a*a*(a√2) =<br /><br />(a^3)(√2)<br /><br />And:<br /><br />h(a^2 + b^2) =<br /><br />h(c^2) = [(a√2)/2]*[(a√2)^2] =<br /><br />[(a√2)/2]*(2a^2) = <br /><br />(a^3)(√2)<br /><br />Notice that, in a 45-45-90 situation, quantities A and B of the original problem are equal. <b>But since we don&#39;t know whether the triangle we&#39;re looking at is that kind of triangle,</b> we don&#39;t have the information we need to determine the answer. If lengths a and b are unequal, Quantities A and B will be unequal.<br /><br />So I&#39;m sticking with (D).Kevin Kimhttps://www.blogger.com/profile/01328790917314282058noreply@blogger.com