To find the [tex]\(y\)[/tex] value for the point located [tex]\(\frac{3}{5}\)[/tex] the distance from [tex]\(L(0, 1)\)[/tex] to [tex]\(M(2, 8)\)[/tex], we need to understand the concept of a point partitioning a line segment in a given ratio.
Here are the steps:
1. Identify coordinates of points L and M:
- Point [tex]\(L\)[/tex] has coordinates [tex]\((0, 1)\)[/tex].
- Point [tex]\(M\)[/tex] has coordinates [tex]\((2, 8)\)[/tex].
2. Determine the fraction of the distance:
- The point is [tex]\(\frac{3}{5}\)[/tex] of the way from [tex]\(L\)[/tex] to [tex]\(M\)[/tex].
3. Calculate the change in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] from [tex]\(L\)[/tex] to [tex]\(M\)[/tex]:
- For [tex]\(L\)[/tex] to [tex]\(M\)[/tex], the change in [tex]\(x\)[/tex] is [tex]\( x_M - x_L = 2 - 0 = 2 \)[/tex].
- For [tex]\(L\)[/tex] to [tex]\(M\)[/tex], the change in [tex]\(y\)[/tex] is [tex]\( y_M - y_L = 8 - 1 = 7 \)[/tex].
4. Determine the coordinates of the point that is [tex]\(\frac{3}{5}\)[/tex] the distance from [tex]\(L\)[/tex] to [tex]\(M\)[/tex]:
- The [tex]\(x\)[/tex]-coordinate of the new point is [tex]\( x_L + \frac{3}{5} \times (x_M - x_L) \)[/tex]:
[tex]\[
x_{\text{new}} = 0 + \frac{3}{5} \times 2 = \frac{6}{5} = 1.2
\][/tex]
- The [tex]\(y\)[/tex]-coordinate of the new point is [tex]\( y_L + \frac{3}{5} \times (y_M - y_L) \)[/tex]:
[tex]\[
y_{\text{new}} = 1 + \frac{3}{5} \times 7 = 1 + \frac{21}{5} = 1 + 4.2 = 5.2
\][/tex]
Thus, the [tex]\(y\)[/tex] value for the point located [tex]\(\frac{3}{5}\)[/tex] the distance from [tex]\(L\)[/tex] to [tex]\(M\)[/tex] is [tex]\(5.2\)[/tex].
Given the choices:
- [tex]\(6\)[/tex]
- [tex]\(5.2\)[/tex]
- [tex]\(3.5\)[/tex]
- [tex]\(4.8\)[/tex]
- [tex]\(1.6\)[/tex]
The correct answer is [tex]\(\boxed{5.2}\)[/tex].
