Looking for reliable answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Explore thousands of questions and answers from a knowledgeable community of experts on our user-friendly platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To find the coordinates of point [tex]\( P \)[/tex] on the directed line segment from point [tex]\( A \)[/tex] to point [tex]\( B \)[/tex], such that [tex]\( P \)[/tex] is [tex]\(\frac{1}{4} \)[/tex] the length of the line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex], you can use the section formula for a given ratio.
The section formula to find the coordinates of a point that divides a line segment joining two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[ (x, y) = \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right) \][/tex]
Here, the points are:
- [tex]\( A = \left(\frac{-29}{4}, \frac{-3}{2}\right) \)[/tex]
- [tex]\( B = \left(\frac{25}{4}, \frac{-1}{2}\right) \)[/tex]
- The ratio in which point [tex]\( P \)[/tex] divides the segment [tex]\( AB \)[/tex] is [tex]\(\frac{1}{4} \)[/tex], so [tex]\( m = 1 \)[/tex] and [tex]\( n = 4 - 1 = 3 \)[/tex].
Using the section formula, plug in the values:
[tex]\[ P_x = \frac{1 \cdot \frac{25}{4} + 3 \cdot \frac{-29}{4}}{1 + 3} \][/tex]
[tex]\[ P_x = \frac{\frac{25}{4} + \frac{-87}{4}}{4} \][/tex]
[tex]\[ P_x = \frac{\frac{25 - 87}{4}}{4} \][/tex]
[tex]\[ P_x = \frac{\frac{-62}{4}}{4} \][/tex]
[tex]\[ P_x = \frac{-62}{16} \][/tex]
[tex]\[ P_x = -\frac{31}{8} \][/tex]
[tex]\[ P_x = -3.875 \][/tex]
Now for the [tex]\( y \)[/tex]-coordinate:
[tex]\[ P_y = \frac{1 \cdot \left(\frac{-1}{2}\right) + 3 \cdot \left(\frac{-3}{2}\right)}{1 + 3} \][/tex]
[tex]\[ P_y = \frac{\frac{-1}{2} + \frac{-9}{2}}{4} \][/tex]
[tex]\[ P_y = \frac{\frac{-10}{2}}{4} \][/tex]
[tex]\[ P_y = \frac{-10}{8} \][/tex]
[tex]\[ P_y = -1.25 \][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] are [tex]\( \left( -3.875, -1.25 \right) \)[/tex].
So, the coordinates of point [tex]\( P \)[/tex] are [tex]\( \left( -3.875, -1.25 \right) \)[/tex].
The section formula to find the coordinates of a point that divides a line segment joining two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[ (x, y) = \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right) \][/tex]
Here, the points are:
- [tex]\( A = \left(\frac{-29}{4}, \frac{-3}{2}\right) \)[/tex]
- [tex]\( B = \left(\frac{25}{4}, \frac{-1}{2}\right) \)[/tex]
- The ratio in which point [tex]\( P \)[/tex] divides the segment [tex]\( AB \)[/tex] is [tex]\(\frac{1}{4} \)[/tex], so [tex]\( m = 1 \)[/tex] and [tex]\( n = 4 - 1 = 3 \)[/tex].
Using the section formula, plug in the values:
[tex]\[ P_x = \frac{1 \cdot \frac{25}{4} + 3 \cdot \frac{-29}{4}}{1 + 3} \][/tex]
[tex]\[ P_x = \frac{\frac{25}{4} + \frac{-87}{4}}{4} \][/tex]
[tex]\[ P_x = \frac{\frac{25 - 87}{4}}{4} \][/tex]
[tex]\[ P_x = \frac{\frac{-62}{4}}{4} \][/tex]
[tex]\[ P_x = \frac{-62}{16} \][/tex]
[tex]\[ P_x = -\frac{31}{8} \][/tex]
[tex]\[ P_x = -3.875 \][/tex]
Now for the [tex]\( y \)[/tex]-coordinate:
[tex]\[ P_y = \frac{1 \cdot \left(\frac{-1}{2}\right) + 3 \cdot \left(\frac{-3}{2}\right)}{1 + 3} \][/tex]
[tex]\[ P_y = \frac{\frac{-1}{2} + \frac{-9}{2}}{4} \][/tex]
[tex]\[ P_y = \frac{\frac{-10}{2}}{4} \][/tex]
[tex]\[ P_y = \frac{-10}{8} \][/tex]
[tex]\[ P_y = -1.25 \][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] are [tex]\( \left( -3.875, -1.25 \right) \)[/tex].
So, the coordinates of point [tex]\( P \)[/tex] are [tex]\( \left( -3.875, -1.25 \right) \)[/tex].
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.