1) C (9,-3.5)
2) C (17,-1.5)
Step-by-step explanation:
1)
To solve this problem, we must divide the segment AB into 8 equal intervals, and then find the point sitting at 3/8 of the whole segment.
The end points of the segment in this problem are:
[tex]A(3,-5)[/tex]
and
[tex]B(19,-1)[/tex]
This means that the x- and y-coordinates of point C are given by the equations:
[tex]x_c=x_a + 3\frac{x_b-x_a}{8}\\y_c=y_a+3\frac{y_b-y_a}{8}[/tex]
And substituting the values of the coordinates of A and B, we find:
[tex]x_c = x_a + 3 \frac{19-3}{8}=3+3\cdot 2 =9\\y_x = y_a + 3 \frac{-1-(-5)}{8}=-5+3\cdot 0.5 =-3.5[/tex]
2)
In this problem, we want to find the coordinates of point C such that:
[tex]\frac{CB}{AC}=\frac{1}{7}[/tex] (1)
As before, the coordinates of the endpoints of the segment AB are:
[tex]A(3,-5)[/tex]
and
[tex]B(19,-1)[/tex]
We can call the coordinates of point C as follows:
[tex]C(x_c,y_c)[/tex]
To satisfy eq.(1) for the x-coordinate, we have:
[tex]\frac{x_b-x_c}{x_c-x_a}=\frac{1}{7}[/tex]
Therefore, by substitution we find:
[tex]\frac{19-x_c}{x_c-3}=\frac{1}{7}\\7(19-x_c)=x_c-3\\8x_c=136 \rightarrow x_c = 17[/tex]
Similarly on the y-coordinate we find:
[tex]\frac{y_b-y_c}{y_c-y_a}=\frac{1}{7}[/tex]
And solving we get:
[tex]\frac{-1-y_c}{y_c-(-5)}=\frac{1}{7}\\7(-1-y_c)=y_c+5\\8y_c=-12 \rightarrow y_c = -1.5[/tex]
