To find the [tex]\( y \)[/tex]-coordinate of the point [tex]\( P \)[/tex] that is [tex]\(\frac{3}{4}\)[/tex] the distance from point [tex]\( E \)[/tex] to point [tex]\( F \)[/tex], follow these steps:
1. Identify the coordinates of points [tex]\( E \)[/tex] and [tex]\( F \)[/tex]:
- Point [tex]\( E \)[/tex] has coordinates [tex]\((2, -3)\)[/tex].
- Point [tex]\( F \)[/tex] has coordinates [tex]\((-2, -1)\)[/tex].
2. Set up the formula to find the coordinates of the point [tex]\( P \)[/tex] which is [tex]\(\frac{3}{4}\)[/tex] of the way from [tex]\( E \)[/tex] to [tex]\( F \)[/tex]:
- For the [tex]\( y \)[/tex]-coordinate, we use the formula:
[tex]\[
y_P = y_1 + \frac{3}{4} \times (y_2 - y_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] are the coordinates of point [tex]\( E \)[/tex] and [tex]\((x_2, y_2)\)[/tex] are the coordinates of point [tex]\( F \)[/tex].
3. Substitute the given coordinates into the formula:
- [tex]\( y_1 = -3 \)[/tex] (coordinate of [tex]\( E \)[/tex])
- [tex]\( y_2 = -1 \)[/tex] (coordinate of [tex]\( F \)[/tex])
Plug these values into the formula:
[tex]\[
y_P = -3 + \frac{3}{4} \times (-1 - (-3))
\][/tex]
4. Simplify the expression inside the parentheses:
[tex]\[
-1 - (-3) = -1 + 3 = 2
\][/tex]
So, the equation now becomes:
[tex]\[
y_P = -3 + \frac{3}{4} \times 2
\][/tex]
5. Calculate the term involving the fraction:
[tex]\[
\frac{3}{4} \times 2 = \frac{3 \times 2}{4} = \frac{6}{4} = 1.5
\][/tex]
6. Add this result to [tex]\( -3 \)[/tex]:
[tex]\[
y_P = -3 + 1.5
\][/tex]
7. Calculate the final value:
[tex]\[
y_P = -1.5
\][/tex]
Thus, the [tex]\( y \)[/tex]-coordinate of the point [tex]\( P \)[/tex] that is [tex]\(\frac{3}{4}\)[/tex] the distance from point [tex]\( E \)[/tex] to point [tex]\( F \)[/tex] is [tex]\(-1.5\)[/tex].
The correct answer is:
[tex]\[
\boxed{-1.5}
\][/tex]
