Discover the answers you need at Westonci.ca, a dynamic Q&A platform where knowledge is shared freely by a community of experts. Explore our Q&A platform to find in-depth answers from a wide range of experts in different fields. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To find the coordinates of point [tex]\( P \)[/tex] that partitions line segment [tex]\( AB \)[/tex] into a part-to-whole ratio of [tex]\( 1:5 \)[/tex], we can use the section formula for internal division.
Given the endpoints [tex]\( A(-9, 3) \)[/tex] and [tex]\( B(1, 8) \)[/tex], and the ratio [tex]\( 1:5 \)[/tex], let's identify the coordinates of point [tex]\( P \)[/tex].
The section formula states:
[tex]\[ P = \left(\frac{m \cdot x_B + n \cdot x_A}{m + n}, \frac{m \cdot y_B + n \cdot y_A}{m + n}\right) \][/tex]
where [tex]\( (x_A, y_A) = (-9, 3) \)[/tex], [tex]\( (x_B, y_B) = (1, 8) \)[/tex], and the ratio [tex]\( m:n = 1:5 \)[/tex].
Let's fill in these values step-by-step:
1. Identify the coordinates and the ratio:
[tex]\[ A = (-9, 3) \][/tex]
[tex]\[ B = (1, 8) \][/tex]
[tex]\[ m = 1, \, n = 5 \][/tex]
2. Formula for the x-coordinate of [tex]\( P \)[/tex]:
[tex]\[ P_x = \frac{m \cdot x_B + n \cdot x_A}{m + n} = \frac{1 \cdot 1 + 5 \cdot -9}{1 + 5} \][/tex]
Now we calculate:
[tex]\[ P_x = \frac{1 \cdot 1 + 5 \cdot -9}{1 + 5} = \frac{1 - 45}{6} = \frac{-44}{6} = -\frac{22}{3} \approx -7.333333333333333 \][/tex]
3. Formula for the y-coordinate of [tex]\( P \)[/tex]:
[tex]\[ P_y = \frac{m \cdot y_B + n \cdot y_A}{m + n} = \frac{1 \cdot 8 + 5 \cdot 3}{1 + 5} \][/tex]
Now we calculate:
[tex]\[ P_y = \frac{1 \cdot 8 + 5 \cdot 3}{1 + 5} = \frac{8 + 15}{6} = \frac{23}{6} \approx 3.8333333333333335 \][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] are [tex]\( \left( -7.333333333333333, 3.8333333333333335 \right) \)[/tex].
So, filling in the correct values for the formula:
[tex]\[ P = \left( \frac{1 \cdot 1 + 5 \cdot (-9)}{1 + 5}, \frac{1 \cdot 8 + 5 \cdot 3}{1 + 5} \right) = \left( \frac{1 + (-45)}{6}, \frac{8 + 15}{6} \right) = \left( -7.333333333333333, 3.8333333333333335 \right) \][/tex]
Finally, the filled formula should be:
[tex]\[ P = \left (\frac{1 \cdot 1 + 5(-9)}{1 + 5}, \frac{1 \cdot 8 + 5(3)}{1 + 5} \right) \][/tex]
Given the endpoints [tex]\( A(-9, 3) \)[/tex] and [tex]\( B(1, 8) \)[/tex], and the ratio [tex]\( 1:5 \)[/tex], let's identify the coordinates of point [tex]\( P \)[/tex].
The section formula states:
[tex]\[ P = \left(\frac{m \cdot x_B + n \cdot x_A}{m + n}, \frac{m \cdot y_B + n \cdot y_A}{m + n}\right) \][/tex]
where [tex]\( (x_A, y_A) = (-9, 3) \)[/tex], [tex]\( (x_B, y_B) = (1, 8) \)[/tex], and the ratio [tex]\( m:n = 1:5 \)[/tex].
Let's fill in these values step-by-step:
1. Identify the coordinates and the ratio:
[tex]\[ A = (-9, 3) \][/tex]
[tex]\[ B = (1, 8) \][/tex]
[tex]\[ m = 1, \, n = 5 \][/tex]
2. Formula for the x-coordinate of [tex]\( P \)[/tex]:
[tex]\[ P_x = \frac{m \cdot x_B + n \cdot x_A}{m + n} = \frac{1 \cdot 1 + 5 \cdot -9}{1 + 5} \][/tex]
Now we calculate:
[tex]\[ P_x = \frac{1 \cdot 1 + 5 \cdot -9}{1 + 5} = \frac{1 - 45}{6} = \frac{-44}{6} = -\frac{22}{3} \approx -7.333333333333333 \][/tex]
3. Formula for the y-coordinate of [tex]\( P \)[/tex]:
[tex]\[ P_y = \frac{m \cdot y_B + n \cdot y_A}{m + n} = \frac{1 \cdot 8 + 5 \cdot 3}{1 + 5} \][/tex]
Now we calculate:
[tex]\[ P_y = \frac{1 \cdot 8 + 5 \cdot 3}{1 + 5} = \frac{8 + 15}{6} = \frac{23}{6} \approx 3.8333333333333335 \][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] are [tex]\( \left( -7.333333333333333, 3.8333333333333335 \right) \)[/tex].
So, filling in the correct values for the formula:
[tex]\[ P = \left( \frac{1 \cdot 1 + 5 \cdot (-9)}{1 + 5}, \frac{1 \cdot 8 + 5 \cdot 3}{1 + 5} \right) = \left( \frac{1 + (-45)}{6}, \frac{8 + 15}{6} \right) = \left( -7.333333333333333, 3.8333333333333335 \right) \][/tex]
Finally, the filled formula should be:
[tex]\[ P = \left (\frac{1 \cdot 1 + 5(-9)}{1 + 5}, \frac{1 \cdot 8 + 5(3)}{1 + 5} \right) \][/tex]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
