To find the [tex]\( y \)[/tex]-coordinate of the point located [tex]\(\frac{3}{5}\)[/tex] of the distance from [tex]\( L(0,1) \)[/tex] to [tex]\( M(2,8) \)[/tex], follow these steps:
1. Determine the coordinates of points [tex]\( L \)[/tex] and [tex]\( M \)[/tex]:
- Point [tex]\( L \)[/tex] has coordinates [tex]\( (0, 1) \)[/tex].
- Point [tex]\( M \)[/tex] has coordinates [tex]\( (2, 8) \)[/tex].
2. Identify the ratio of the distance:
- The point of interest is [tex]\(\frac{3}{5}\)[/tex] of the way from [tex]\( L \)[/tex] to [tex]\( M \)[/tex].
3. Calculate the difference in the [tex]\( y \)[/tex]-coordinates between [tex]\( L \)[/tex] and [tex]\( M \)[/tex]:
- The change in [tex]\( y \)[/tex]-coordinates ([tex]\(\Delta y\)[/tex]) from [tex]\( L \)[/tex] to [tex]\( M \)[/tex] is [tex]\( 8 - 1 = 7 \)[/tex].
4. Calculate the [tex]\( y \)[/tex]-coordinate for the point [tex]\(\frac{3}{5}\)[/tex] of the way from [tex]\( L \)[/tex] to [tex]\( M \)[/tex]:
- The [tex]\( y \)[/tex]-coordinate of this point will be the [tex]\( y \)[/tex]-coordinate of [tex]\( L \)[/tex] plus [tex]\(\frac{3}{5}\)[/tex] of the change in [tex]\( y \)[/tex]-coordinates.
- So, [tex]\[
y = y_L + \frac{3}{5} \cdot (y_M - y_L)
\][/tex]
- Substituting in the values, we get:
[tex]\[
y = 1 + \frac{3}{5} \cdot (8 - 1)
\][/tex]
- Simplify the expression within the parentheses:
[tex]\[
y = 1 + \frac{3}{5} \cdot 7
\][/tex]
- Calculate [tex]\(\frac{3}{5} \cdot 7\)[/tex]:
[tex]\[
\frac{3}{5} \cdot 7 = \frac{21}{5} = 4.2
\][/tex]
- Add this value to the initial [tex]\( y \)[/tex]-coordinate:
[tex]\[
y = 1 + 4.2 = 5.2
\][/tex]
Hence, the [tex]\( y \)[/tex]-coordinate of the point located [tex]\(\frac{3}{5}\)[/tex] the distance from [tex]\( L \)[/tex] to [tex]\( M \)[/tex] is [tex]\( 5.2 \)[/tex].
