Let's find the coordinates of point [tex]\( P \)[/tex] that partitions line segment [tex]\( AB \)[/tex] in a part-to-whole ratio of 1:5.
First, let us recall the Section Formula: If a point [tex]\( P(x, y) \)[/tex] divides the line joining two points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] in the ratio [tex]\( m : n \)[/tex], then the coordinates of [tex]\( P \)[/tex] are given by:
[tex]\[ P_x = \frac{mx_2 + nx_1}{m+n} \][/tex]
[tex]\[ P_y = \frac{my_2 + ny_1}{m+n} \][/tex]
However, since we are given a part-to-whole ratio instead of a fractional ratio directly, we need to subtract the part from the whole to get the other segment ratio value:
Let:
- [tex]\( A = (-9, 3) \)[/tex]
- [tex]\( B = (1, 8) \)[/tex]
- Ratio [tex]\( m : (m+n) = 1 : 5 \)[/tex]
We can extract the ratio values:
- [tex]\( m = 1 \)[/tex]
- [tex]\( n = 5 - 1 = 4 \)[/tex]
Plugging the values into the section formula:
[tex]\[ P_x = \frac{m x_2 + n x_1}{m + n} = \frac{1 \cdot 1 + 4 \cdot (-9)}{1 + 4} \][/tex]
[tex]\[ P_y = \frac{m y_2 + n y_1}{m + n} = \frac{1 \cdot 8 + 4 \cdot 3}{1 + 4} \][/tex]
Let's compute these individually to fill in the equation given:
For [tex]\( P_x \)[/tex]:
[tex]\[ P_x = -9 + \left( \frac{1}{5} \right) \left( 1 + 9 \right) \][/tex]
For [tex]\( P_y \)[/tex]:
[tex]\[ P_y = 3 + \left( \frac{1}{5} \right) \left( 8 - 3 \right) \][/tex]
So, the detailed values for the formula [tex]\( P = \left( P_x, P_y \right) \)[/tex] should be filled as follows:
[tex]\[ P = \left( -9 + \frac{1}{5}(1 - (-9)), 3 + \frac{1}{5}(8 - 3) \right) \][/tex]
Here are the correct values to fill in the formula:
[tex]\[ P = \left( -9 + \frac{1}{5}(1 -(-9)), 3 + \frac{1}{5}(8 - 3) \right) \][/tex]
Given this information, the specific step-by-step calculations leading to [tex]\( P = (-7.0, 4.0) \)[/tex] should fill in the blanks as follows:
[tex]\[ P = \left( -9 + \frac{1}{5}(1 + 9), 3 + \frac{1}{5}(8 - 3) \right) \][/tex]
