To find the [tex]$y$[/tex]-coordinate of a point that divides the horizontal side in the ratio [tex]$2:3$[/tex], we use the given formula from the lesson:
[tex]\[ y_C = \frac{a y_a + b y_A}{a + b} \][/tex]
Here, the ratio [tex]\( a : b \)[/tex] is given as [tex]\( 2 : 3 \)[/tex]. This means:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 3 \)[/tex]
We also need to know the [tex]$y$[/tex]-coordinates of the two points between which the division occurs. Let's denote these coordinates as [tex]\( y_a \)[/tex] and [tex]\( y_A \)[/tex]:
- [tex]\( y_a = 0 \)[/tex] (the [tex]\( y \)[/tex]-coordinate of the first point)
- [tex]\( y_A = 10 \)[/tex] (the [tex]\( y \)[/tex]-coordinate of the second point)
Now we substitute these values into the formula:
[tex]\[ y_C = \frac{(2 \cdot 0) + (3 \cdot 10)}{2 + 3} \][/tex]
Evaluating the numerator and the denominator separately:
1. For the numerator:
[tex]\[ 2 \cdot 0 + 3 \cdot 10 = 0 + 30 = 30 \][/tex]
2. For the denominator:
[tex]\[ 2 + 3 = 5 \][/tex]
Now we divide the numerator by the denominator:
[tex]\[ y_C = \frac{30}{5} = 6.0 \][/tex]
So, the [tex]$y$[/tex]-coordinate of the point that divides the horizontal side in the ratio [tex]$2:3$[/tex] is:
[tex]\[ y_C = 6.0 \][/tex]
