Answer :
Let's determine the [tex]\( y \)[/tex]-coordinate for the point that is [tex]\(\frac{3}{4}\)[/tex] of the way from point [tex]\( E \)[/tex] to point [tex]\( F \)[/tex].
### Step-by-Step Solution:
1. Identify the Coordinates of Points [tex]\( E \)[/tex] and [tex]\( F \)[/tex]:
- [tex]\( E = (2, -3) \)[/tex]
- [tex]\( F = (-2, -1) \)[/tex]
2. Determine the Change in [tex]\( y \)[/tex]-coordinates from [tex]\( E \)[/tex] to [tex]\( F \)[/tex]:
- Change in [tex]\( y \)[/tex] ([tex]\(\Delta y\)[/tex]) = [tex]\( y_F - y_E = -1 - (-3) \)[/tex]
- Simplify the expression:
[tex]\[ \Delta y = -1 + 3 = 2 \][/tex]
3. Calculate the [tex]\( y \)[/tex]-coordinate for the Point [tex]\(\frac{3}{4}\)[/tex] the Distance from [tex]\( E \)[/tex] to [tex]\( F \)[/tex]:
- We need to move [tex]\(\frac{3}{4}\)[/tex] of the [tex]\(\Delta y\)[/tex] from [tex]\( y_E \)[/tex]:
- The ratio [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[ \Delta y \times \frac{3}{4} = 2 \times \frac{3}{4} = 1.5 \][/tex]
- Add this to the starting [tex]\( y \)[/tex]-coordinate of point [tex]\( E \)[/tex]:
[tex]\[ y = y_E + 1.5 = -3 + 1.5 = -1.5 \][/tex]
By completing these calculations, we conclude that the [tex]\( y \)[/tex]-coordinate for the point that is [tex]\(\frac{3}{4}\)[/tex] the distance from [tex]\( E \)[/tex] to [tex]\( F \)[/tex] is:
[tex]\[ \boxed{-1.5} \][/tex]
### Step-by-Step Solution:
1. Identify the Coordinates of Points [tex]\( E \)[/tex] and [tex]\( F \)[/tex]:
- [tex]\( E = (2, -3) \)[/tex]
- [tex]\( F = (-2, -1) \)[/tex]
2. Determine the Change in [tex]\( y \)[/tex]-coordinates from [tex]\( E \)[/tex] to [tex]\( F \)[/tex]:
- Change in [tex]\( y \)[/tex] ([tex]\(\Delta y\)[/tex]) = [tex]\( y_F - y_E = -1 - (-3) \)[/tex]
- Simplify the expression:
[tex]\[ \Delta y = -1 + 3 = 2 \][/tex]
3. Calculate the [tex]\( y \)[/tex]-coordinate for the Point [tex]\(\frac{3}{4}\)[/tex] the Distance from [tex]\( E \)[/tex] to [tex]\( F \)[/tex]:
- We need to move [tex]\(\frac{3}{4}\)[/tex] of the [tex]\(\Delta y\)[/tex] from [tex]\( y_E \)[/tex]:
- The ratio [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[ \Delta y \times \frac{3}{4} = 2 \times \frac{3}{4} = 1.5 \][/tex]
- Add this to the starting [tex]\( y \)[/tex]-coordinate of point [tex]\( E \)[/tex]:
[tex]\[ y = y_E + 1.5 = -3 + 1.5 = -1.5 \][/tex]
By completing these calculations, we conclude that the [tex]\( y \)[/tex]-coordinate for the point that is [tex]\(\frac{3}{4}\)[/tex] the distance from [tex]\( E \)[/tex] to [tex]\( F \)[/tex] is:
[tex]\[ \boxed{-1.5} \][/tex]
