ANSWERS
a. Coodri (2, 6)
b. 8√5
c. -2
d. y = -2x + 2
EXPLANATION
a. To find the coordinate of point P we can do a diagram first that will help us see the problem more clearly:
U is the midpoint of segment PT, which means that the horizontal distance between U and T is the same as between P and U, but in the opposite direction (in this case to the left of U). The same happens to the vertical distance. Therefore, we have that the horizontal distance is:
[tex]x_T-x_U=10-6=4_{}[/tex]
Therefore the x-coordinate of point P is:
[tex]x_P=x_U-4=6-4=2_{}_{}[/tex]
The vertical distance between points U and T is:
[tex]y_T-y_U=-2-(-10)=-2+10=8_{}[/tex]
So the y-coordinate of point P is:
[tex]y_P=y_U+8=-2+8=6[/tex]
Therefore the coordinates of point P are (2, 6)
b. The length of a segment with endpoints (x1, y1) and (x2, y2) is:
[tex]d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
In this case the endpoints are P (2, 6) and T (10, -10). The length of the segment is:
[tex]\begin{gathered} d=\sqrt[]{(10-2)^2+(-10-6)^2} \\ d=\sqrt[]{8^2+16^2} \\ d=\sqrt[]{64+256} \\ d=\sqrt[]{320} \\ d=8\sqrt[]{5} \end{gathered}[/tex]
c. The slope between two points (x1, y1) and (x2, y2) is:
[tex]m=\frac{y_1-y_2}{x_1-x_2}[/tex]
In this case the points we can use are P and T, or P and U or U and T. This is because they are all collinear, so between each pair of points the slope is the same. We'll use points P(2, 6) and T(10, -10):
[tex]m=\frac{-10-6}{10-2}=\frac{-16}{8}=-2[/tex]
The slope is -2.
d. First we have to find the equation of the line with the segment PT. It has the form:
[tex]y=mx+b[/tex]
We have m, which is the slope and we need b, which is the y-intercept. The equation we have for now is:
[tex]y=-2x+b[/tex]
To find b we just have to replace (x, y) by any point on the line and solve for b. Using point U(6, -2):
[tex]\begin{gathered} -2=-2\cdot6+b \\ -2=-12+b \\ b=12-2 \\ b=10 \end{gathered}[/tex]
The equation of the line with segment PT is:
[tex]y=-2x+10[/tex]
But we have to find the equation of a parallel line to this one. This is very easy now, because parallel lines have the same slope, but different y-intercept. Hence, the equation of a parallel line would be the same equation changing the y-intercept 10 by any arbitrary value. Let's say the y-intercept is 2, so a parallel line is:
[tex]y=-2x+2[/tex]
