To solve for the [tex]\( x \)[/tex]-coordinate of a point [tex]\( C \)[/tex] that divides the horizontal side in the ratio [tex]\( 2:3 \)[/tex], we will use the given formula:
[tex]\[ x_C = \frac{a x_a + b z_A}{a + b} \][/tex]
Here, the following values are provided:
- [tex]\( a = 2 \)[/tex] (the first part of the ratio)
- [tex]\( b = 3 \)[/tex] (the second part of the ratio)
- [tex]\( x_a = 1 \)[/tex] (the [tex]\( x \)[/tex]-coordinate of point [tex]\( A \)[/tex])
- [tex]\( z_A = 4 \)[/tex] (the [tex]\( z \)[/tex]-coordinate of point [tex]\( B \)[/tex])
Now we substitute these values into the formula:
[tex]\[ x_C = \frac{(2 \times 1) + (3 \times 4)}{2 + 3} \][/tex]
Let's break this down step by step:
1. Calculate the product of [tex]\( a \)[/tex] and [tex]\( x_a \)[/tex]:
[tex]\[ 2 \times 1 = 2 \][/tex]
2. Calculate the product of [tex]\( b \)[/tex] and [tex]\( z_A \)[/tex]:
[tex]\[ 3 \times 4 = 12 \][/tex]
3. Add the results of these products:
[tex]\[ 2 + 12 = 14 \][/tex]
4. Sum the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ 2 + 3 = 5 \][/tex]
5. Finally, divide the sum of the products by the sum of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x_C = \frac{14}{5} = 2.8 \][/tex]
Therefore, the [tex]\( x \)[/tex]-coordinate of the point [tex]\( C \)[/tex] that divides the horizontal side in the ratio [tex]\( 2:3 \)[/tex] is [tex]\( 2.8 \)[/tex].
