Question 332266
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(a).  Use the two-point form of an equation of a line:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ -\ y_1\ =\ \left(\frac{y_1\ -\ y_2}{x_1\ -\ x_2}\right)(x\ -\ x_1) ]


where *[tex \Large \left(x_1,y_1\right)] and *[tex \Large \left(x_2,y_2\right)] are the coordinates of points A and C.


(b).  Step 1.  Use the


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{y_1\ -\ y_2}{x_1\ -\ x_2} ]


portion of the two-point form to determine the slope of the line containing segment AB.


Step 2:  Calculate the negative reciprocal of the slope determined in Step 1 because:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  L_1\ \perp\ L_2 \ \ \Leftrightarrow\ \ m_1\ =\ -\frac{1}{m_2}\ \text{ and } m_1,\, m_2\, \neq\, 0]


Step 3:  Use the point-slope form of an equation of a line:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ -\ y_1\ =\ m(x\ -\ x_1) ]


where *[tex \Large \left(x_1,y_1\right)] are the coordinates of point C and *[tex \Large m] is the slope calculated in (b) Step 2.


(c) Step 1: Use the mid-point formulas:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x_m\ = \frac{x_1 + x_2}{2}] and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y_m\ = \frac{y_1 + y_2}{2}]


where *[tex \Large \left(x_1,y_1\right)] and *[tex \Large \left(x_2,y_2\right)] are the coordinates of points A and C to calculate the midpoint *[tex \LARGE (x_m,y_m)] of segment AC.


Use the slope calculated in (b) Step 2 and the midpoint calculated in (c) Step 1 with the point-slope form to derive the equation of the perpendicular bi-sector of AC.


(d)  Step 1:  Use the distance formula 3 times:


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


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ d\ =\ sqrt{(x_1\ -\ x_3)^2\ +\ (y_1\ -\ y_3)^2}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ d\ =\ sqrt{(x_2\ -\ x_3)^2\ +\ (y_2\ -\ y_3)^2}]


where *[tex \Large \left(x_1,y_1\right)], *[tex \Large \left(x_2,y_2\right)], and *[tex \Large \left(x_3,y_3\right)] are the coordinates of points A, B, and C  .


Step 2: Sum the three results.



John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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