<p>In the xy-plane, point R (2,3) and point S (5,6) are two vertices of triangle RST. If the sum of the slopes of the sides of the sides of the triangle is 1, which of the following angles could be a right angle?</p>
<pre><code> I. angle R
II. angle S
III. angle T
</code></pre>
<p>a. none
b. I only
c. III only
d. I and II only
e. I, II, and III only </p>
<p>The correct answer is a.</p>
<hr>
<p>For all the positive values of x, the function f is defined by f(x)= x^2 minus X^-2. Of the following, which is the best approximation of f(x) for values of x greater than 1,000?</p>
<p>a.x^2
b.x^4
c.x^5
d.x^6
e.x^9</p>
<p>The correct answer is a.</p>
<p>1) The slope of RS = 1, so if we assume that there is a right angle, then the slope of another line in that triangle will be the opposite reciprocal, so -1. And since 1 + (-1) + x = 1 then x has to = 1. Well that does not work, because then we have parallel lines in a triangle, and that does not work. So it has to be none of them.</p>
<p>2) we have x^2 - (1/(x^2))</p>
<p>As x increases to, lets say one billion, the (1/(x^2)) becomes smaller and smaller. At a billion it is pretty close to zero. While x^2 is getting bigger and bigger. So basically, the larger the number, the more important x^2 becomes, and the less important (1/(x^2)) becomes. So if you choose an arbitrarily large number, like 1,000, the value of f(1000) in the given function, will look allot like f(1000) in an x squared function.</p>
<p>Hope that makes sense.</p>
<ol>
<li>The above proves that the right angle is not at R or S. Slope of RS is 1 and if slope of RT or ST is -1, the other one would need to be parallel to RS with slope of 1.</li>
</ol>
<p>To eliminate the possibility of angle T being a right angle, If T was the right angle, the slopes would be
RS -> 1
ST -> x
RT -> -1/x (since RT would be perpendicular to ST & product of
their slopes would be -1)</p>
<p>Then the sum of the slopes would be 1+x+(-1/x) = 1
=> x+(-1/x) = 0
=> x^2 = 1
=> x = 1 or -1
=> -1/x = -1 or 1
So, the slopes would be 1,1,-1 in all cases which cannot be a triangle, since two of the lines would be parallel.</p>