To find the [tex]$y$[/tex]-coordinate of a point that divides the horizontal side in the ratio [tex]$2:3$[/tex], we will use the formula provided in the lesson:
[tex]\[ y_C = \frac{a r + b y}{a + b} \][/tex]
Here:
- [tex]\(a\)[/tex] is the first ratio part.
- [tex]\(b\)[/tex] is the second ratio part.
- [tex]\(r\)[/tex] and [tex]\(y\)[/tex] are the [tex]$y$[/tex]-coordinates of the endpoints of the horizontal side.
In this specific problem:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 3\)[/tex]
- The horizontal side has both its endpoints at [tex]$y=0$[/tex]. Therefore, [tex]\(r = 0\)[/tex] and [tex]\(y = 0\)[/tex].
We can plug these values into the formula:
[tex]\[ y_C = \frac{2 \cdot 0 + 3 \cdot 0}{2 + 3} \][/tex]
Simplify the expression:
[tex]\[ y_C = \frac{0 + 0}{5} \][/tex]
[tex]\[ y_C = \frac{0}{5} \][/tex]
[tex]\[ y_C = 0 \][/tex]
Therefore, the [tex]$y$[/tex]-coordinate of the point that divides the horizontal side in the ratio [tex]$2:3$[/tex] is:
[tex]\[ y_C = 0 \][/tex]
The [tex]$y$[/tex]-coordinate is [tex]\(0.0\)[/tex].
