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Point [tex]\(X\)[/tex] is located at [tex]\((2,-6)\)[/tex], and point [tex]\(Z\)[/tex] is located at [tex]\((0,5)\)[/tex]. Find the [tex]\(y\)[/tex] value for the point [tex]\(Y\)[/tex] that is located [tex]\(\frac{1}{5}\)[/tex] the distance from point [tex]\(X\)[/tex] to point [tex]\(Z\)[/tex].

A. [tex]\(-3.6\)[/tex]
B. [tex]\(-3.8\)[/tex]
C. [tex]\(-4\)[/tex]
D. [tex]\(-4.2\)[/tex]

Sagot :

GinnyAnswer
To find the [tex]$y$[/tex] value for point [tex]\( Y \)[/tex] that is [tex]\(\frac{1}{5}\)[/tex] the distance from point [tex]\(X\)[/tex] to point [tex]\(Z\)[/tex], follow these steps:

1. Identify the coordinates of points [tex]\(X\)[/tex] and [tex]\(Z\)[/tex]:
- Point [tex]\(X\)[/tex] is [tex]\((2, -6)\)[/tex]
- Point [tex]\(Z\)[/tex] is [tex]\((0, 5)\)[/tex]

2. Calculate the difference in the x-coordinates ([tex]\(dx\)[/tex]) and y-coordinates ([tex]\(dy\)[/tex]):
- [tex]\(dx = Z_x - X_x = 0 - 2 = -2\)[/tex]
- [tex]\(dy = Z_y - X_y = 5 - (-6) = 5 + 6 = 11\)[/tex]

3. Calculate the coordinates of point [tex]\(Y\)[/tex] using the distance ratio [tex]\(\frac{1}{5}\)[/tex]:
- The x-coordinate of point [tex]\(Y\)[/tex] is: [tex]\(Y_x = X_x + \frac{1}{5} \times dx = 2 + \frac{1}{5} \times -2 = 2 - 0.4 = 1.6\)[/tex]
- The y-coordinate of point [tex]\(Y\)[/tex] is: [tex]\(Y_y = X_y + \frac{1}{5} \times dy = -6 + \frac{1}{5} \times 11 = -6 + 2.2 = -3.8\)[/tex]

Therefore, the [tex]\(y\)[/tex] value for point [tex]\(Y\)[/tex] is:
[tex]\[ -3.8 \][/tex]
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