Question 549599
<font face="Times New Roman" size="+2">


*[tex \Large x\ =\ 0] is the *[tex \Large x]-axis.  *[tex \Large y\ =\ 0] is the *[tex \Large y]-axis.  *[tex \Large x\ =\ c] is a vertical line through the point *[tex \Large (c,0)] which must be to the right of the *[tex \Large y]-axis because *[tex \Large c] is constrained to the positive numbers.  *[tex \Large y\ =\ kx\ +\ b] is a line with a slope of *[tex \Large k] and a *[tex \Large y]-intercept of *[tex \Large (0,b)].  The line must slope upward to the right and *[tex \Large (0,b)] must be above the *[tex \Large x]-axis because both *[tex \Large k] and *[tex \Large b] are constrained to the positive numbers.


In sum, these four lines bound a quadrilateral with vertices at *[tex \Large (0,0)], *[tex \Large (0,b)], *[tex \Large (c,kc+b)], and *[tex \Large (c,0)].  Note: *[tex \Large kc\ +\ b] comes from substituting the *[tex \Large x] value *[tex \Large c] for *[tex \Large x] in the equation *[tex \Large y\ =\ kx\ +\ b] yielding *[tex \Large y\ =\ kc\ +\ b]


Since *[tex \Large x\ =\ c] is a vertical line it must be parallel to the *[tex \Large y]-axis.  Furthermore, since *[tex \Large k] is presumed to be non-zero, the line described by *[tex \Large y\ =\ kx\ +\ b] is NOT parallel to the *[tex \Large x]-axis.  Therefore, two parallel sides and two not parallel sides -- we have a trapezoid ladies and gents; trapezium if you are in the UK. 


The area of a trapezoid is the average of the measures of the two parallel sides times the measure of the altitude of the trapezoid, i.e. the distance between the two parallel sides.  Since the side with endpoints at the origin and at *[tex \Large (c,0)] is perpendicular to both of the parallel sides, the distance between the two parallel sides is simply the distance from *[tex \Large (0,0)] to *[tex \Large (c,0)], which is simply *[tex \Large c] (You can verify that with the distance formula if you like).


The measure of the shorter parallel side is the distance from *[tex \Large (0,0)] to *[tex \Large (0,b)], i.e. *[tex \Large b].  The measure of the longer side is the distance from *[tex \Large (c,0)] to *[tex \Large (c,kc+b)].  Here's the distance formula for that one if the answer isn't obvious.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ d\ =\ sqrt{(x_1\ -\ x_2)^2\ +\ (y_1\ -\ y_2)^2}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ d\ =\ sqrt{(c\ -\ c)^2\ +\ (0\ -\ (kc+b))^2}\ =\ kc\ +\ b]


So the average of the measures of the two parallel sides is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{b\ +\ (kc\ +\ b)}{2}]


Multiply this by the altitude, *[tex \Large c], to get:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ A=\ \frac{2bc\ +\ kc^2}{2}]



John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
<div style="text-align:center"><a href="http://outcampaign.org/" target="_blank"><img src="http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png" border="0" alt="The Out Campaign: Scarlet Letter of Atheism" width="143" height="122" /></a></div>
</font>