Respuesta :
Answer:
The x-intercept of CD is B(18/5,0). The point C(32,-71) lies on the line CD.
Step-by-step explanation:
Given information: CD is perpendicular bisector of AB.
The coordinates of point A are (-3, 2) and the coordinates of point B are (7, 6).
Midpoint of AB is C.
[tex]C=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})=(\frac{7-3}{2},\frac{2+6}{2})=(2,4)[/tex]
The coordinates of C are (2,4).
The slope of line AB is
[tex]m_1=\frac{y_2-y_1}{x_2-x_1}=\frac{6-2}{7-(-3)}=\frac{4}{10}=\frac{2}{5}[/tex]
The product of slopes of two perpendicular lines is -1. Since the line CD is perpendicular to AB, therefore the slope of CD is
[tex]m_2=-\frac{5}{2}[/tex]
The point slope form of a line is
[tex]y-y_1=m(x-x_1)[/tex]
The slope of line CD is [tex]-\frac{5}{2}[/tex] and the line passing through the point (2,4), the equation of line CD is
[tex]y-4=-\frac{5}{2}(x-2)[/tex]
[tex]y=-\frac{5}{2}x+5+4[/tex]
[tex]y=-\frac{5}{2}x+9[/tex] .... (1)
The equation of CD is [tex]y=-\frac{5}{2}x+9[/tex].
Put y=0, to find the x-intercept.
[tex]0=-\frac{5}{2}x+9[/tex]
[tex]\frac{5}{2}x=9[/tex]
[tex]x=\frac{18}{5}[/tex]
Therefore the x-intercept of CD is B(18/5,0).
Put x=-52 in equation (1).
[tex]y=-\frac{5}{2}(-52)+9=139[/tex]
Put x=-20 in equation (1).
[tex]y=-\frac{5}{2}(-20)+9=59[/tex]
Put x=32 in equation (1).
[tex]y=-\frac{5}{2}(32)+9=-71[/tex]
Put x=-54 in equation (1).
[tex]y=-\frac{5}{2}(-54)+9=144[/tex]
Only point (32,-71) satisfies the equation of CD. Therefore the point C(32,-71) lies on the line CD.
Answer:
The x-intercept of CD is B(18/5,0). The point C(32,-71) lies on the line CD.
Step-by-step explanation:
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