Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Join our Q&A platform and connect with professionals ready to provide precise answers to your questions in various areas. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
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]\(\frac{1}{3}\)[/tex] the length of the line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex], we can use the section formula for internal division.
Given:
- Coordinates of [tex]\( A \)[/tex] are [tex]\( A(1, 5) \)[/tex]
- Coordinates of [tex]\( B \)[/tex] are [tex]\( B(0, 3) \)[/tex]
- The ratio in which [tex]\( P \)[/tex] divides the line segment [tex]\( AB \)[/tex] is [tex]\( m:n = 1:2 \)[/tex]
The section formula states that if a point [tex]\( P \)[/tex] divides a line segment [tex]\( AB \)[/tex] with coordinates [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex], the coordinates of [tex]\( P \)[/tex] are given by:
[tex]\[ P_x = \left( \frac{m}{m+n} \right)(x_2 - x_1) + x_1 \][/tex]
[tex]\[ P_y = \left( \frac{m}{m+n} \right)(y_2 - y_1) + y_1 \][/tex]
Substituting the given values:
- [tex]\( x_1 = 1 \)[/tex]
- [tex]\( y_1 = 5 \)[/tex]
- [tex]\( x_2 = 0 \)[/tex]
- [tex]\( y_2 = 3 \)[/tex]
- [tex]\( m = 1 \)[/tex]
- [tex]\( n = 2 \)[/tex]
First, calculate the [tex]\( x \)[/tex]-coordinate of [tex]\( P \)[/tex]:
[tex]\[ P_x = \left( \frac{1}{1+2} \right)(0 - 1) + 1 \][/tex]
[tex]\[ P_x = \left( \frac{1}{3} \right)(-1) + 1 \][/tex]
[tex]\[ P_x = -\frac{1}{3} + 1 \][/tex]
[tex]\[ P_x = \frac{3}{3} - \frac{1}{3} \][/tex]
[tex]\[ P_x = \frac{2}{3} \][/tex]
Next, calculate the [tex]\( y \)[/tex]-coordinate of [tex]\( P \)[/tex]:
[tex]\[ P_y = \left( \frac{1}{1+2} \right)(3 - 5) + 5 \][/tex]
[tex]\[ P_y = \left( \frac{1}{3} \right)(-2) + 5 \][/tex]
[tex]\[ P_y = -\frac{2}{3} + 5 \][/tex]
[tex]\[ P_y = 5 - \frac{2}{3} \][/tex]
[tex]\[ P_y = \frac{15}{3} - \frac{2}{3} \][/tex]
[tex]\[ P_y = \frac{13}{3} \][/tex]
Thus, the coordinates of point [tex]\( P \)[/tex] are:
[tex]\[ P \left( \frac{2}{3}, \frac{13}{3} \right) \][/tex]
Converted to decimal form, the coordinates are approximately:
[tex]\[ P (0.6666666666666667, 4.333333333333333) \][/tex]
Therefore, the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-coordinates of point [tex]\( P \)[/tex] are [tex]\(\boxed{0.6666666666666667, 4.333333333333333}\)[/tex].
Given:
- Coordinates of [tex]\( A \)[/tex] are [tex]\( A(1, 5) \)[/tex]
- Coordinates of [tex]\( B \)[/tex] are [tex]\( B(0, 3) \)[/tex]
- The ratio in which [tex]\( P \)[/tex] divides the line segment [tex]\( AB \)[/tex] is [tex]\( m:n = 1:2 \)[/tex]
The section formula states that if a point [tex]\( P \)[/tex] divides a line segment [tex]\( AB \)[/tex] with coordinates [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex], the coordinates of [tex]\( P \)[/tex] are given by:
[tex]\[ P_x = \left( \frac{m}{m+n} \right)(x_2 - x_1) + x_1 \][/tex]
[tex]\[ P_y = \left( \frac{m}{m+n} \right)(y_2 - y_1) + y_1 \][/tex]
Substituting the given values:
- [tex]\( x_1 = 1 \)[/tex]
- [tex]\( y_1 = 5 \)[/tex]
- [tex]\( x_2 = 0 \)[/tex]
- [tex]\( y_2 = 3 \)[/tex]
- [tex]\( m = 1 \)[/tex]
- [tex]\( n = 2 \)[/tex]
First, calculate the [tex]\( x \)[/tex]-coordinate of [tex]\( P \)[/tex]:
[tex]\[ P_x = \left( \frac{1}{1+2} \right)(0 - 1) + 1 \][/tex]
[tex]\[ P_x = \left( \frac{1}{3} \right)(-1) + 1 \][/tex]
[tex]\[ P_x = -\frac{1}{3} + 1 \][/tex]
[tex]\[ P_x = \frac{3}{3} - \frac{1}{3} \][/tex]
[tex]\[ P_x = \frac{2}{3} \][/tex]
Next, calculate the [tex]\( y \)[/tex]-coordinate of [tex]\( P \)[/tex]:
[tex]\[ P_y = \left( \frac{1}{1+2} \right)(3 - 5) + 5 \][/tex]
[tex]\[ P_y = \left( \frac{1}{3} \right)(-2) + 5 \][/tex]
[tex]\[ P_y = -\frac{2}{3} + 5 \][/tex]
[tex]\[ P_y = 5 - \frac{2}{3} \][/tex]
[tex]\[ P_y = \frac{15}{3} - \frac{2}{3} \][/tex]
[tex]\[ P_y = \frac{13}{3} \][/tex]
Thus, the coordinates of point [tex]\( P \)[/tex] are:
[tex]\[ P \left( \frac{2}{3}, \frac{13}{3} \right) \][/tex]
Converted to decimal form, the coordinates are approximately:
[tex]\[ P (0.6666666666666667, 4.333333333333333) \][/tex]
Therefore, the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-coordinates of point [tex]\( P \)[/tex] are [tex]\(\boxed{0.6666666666666667, 4.333333333333333}\)[/tex].
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.