<p>I have a problem with #8, page 776 in the Blue Book. The questions goes like this:</p>
<li>Rectangle ABCD lies in the xy-coordinate plane so that its sides are <em>NOT</em> parallel to the axes. What is the product of the slopes of all four sides of rectangle ABCD? </li>
</ol>
<p>(A) -2
(B) -1
(C) 0
(D) 1
(E) 2</p>
<p>The answer was (D).</p>
<p>I understand why A, B, and C are wrong but why can’t the answer be E as well since a square can also be considered a rectangle, right?</p>
<p>*Note: There is no figure or image drawn for this problem.</p>
<p>Squares are rectangles, yes, but that's irrelevant: this is about slope.</p>
<p>AB and CD are parallel to one another. Let's let the slope of each of those line segments is x.</p>
<p>BC and DA are also parallel to one another. They are also perpendicular to AB and CD. That means that the slope of BC and DA is 1/x.</p>
<p>That means that the number you want is x times x times 1/x times 1/x, which is x^2/x^2.</p>
<p>We know that x^2 =/= 0 because the sides of ABCD are not parallel to the axes.</p>
<p>If x^2 =/= 0. then x^2/x^2 = 1.</p>
<p>If ABCD, in addition to being a rectangle, is a square, all that means is that the lengths of the sides are all the same. That tells you nothing about the slope, and in particular it doesn't mean that the product of the slopes is anything other than x^2/x^2.</p>
<p>They are not parallel to axes so the problem can be solved (thats the only reason they tell you this).</p>
<p>Now, a rectangle has perpendicular sides, so that means that their slopes will be opposites (and recipricals, but we can make it easier and assume 1).</p>
<p>If the slopes of 2 sides are 1, then the slopes of the other 2 sides are -1.</p>
<p>(-1)(-1)(1)(1) = 1. </p>
<p>I might not have clarified my method that well.. if you don't get something just ask.</p>