Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Experience the ease of finding accurate answers to your questions from a knowledgeable community of professionals. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To find the coordinates of point [tex]\( P \)[/tex] on the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex], such that [tex]\( P \)[/tex] is [tex]\( \frac{2}{3} \)[/tex] of the length of the segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex], we can use the section formula for internal division.
Given points [tex]\( A(x_1, y_1) = (2, -1) \)[/tex] and [tex]\( B(x_2, y_2) = (4, -3) \)[/tex], let's calculate the coordinates step-by-step:
1. We know that [tex]\( P \)[/tex] divides the segment [tex]\( AB \)[/tex] in the ratio [tex]\( \frac{2}{3} \)[/tex]. Therefore, we can denote [tex]\( m = 2 \)[/tex] and [tex]\( n = 1 \)[/tex]. This comes from the ratio [tex]\( 2 \)[/tex] parts of [tex]\( AB \)[/tex] to [tex]\( 1 \)[/tex] part of [tex]\( AP \)[/tex] added together giving [tex]\( m + n = 3 \)[/tex].
2. Substitute the ratio and coordinates into the section formula:
[tex]\[ x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1 \][/tex]
[tex]\[ y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1 \][/tex]
Plugging in the values:
[tex]\[ x = \left(\frac{2}{3}\right)(4 - 2) + 2 \][/tex]
Now calculate the [tex]\( x \)[/tex]-coordinate:
[tex]\[ x = \left(\frac{2}{3}\right)(2) + 2 = \frac{4}{3} + 2 = \frac{4}{3} + \frac{6}{3} = \frac{10}{3} \approx 3.33 \][/tex]
Now for the [tex]\( y \)[/tex]-coordinate:
[tex]\[ y = \left(\frac{2}{3}\right)(-3 - (-1)) + (-1) \][/tex]
[tex]\[ y = \left(\frac{2}{3}\right)(-2) - 1 = -\frac{4}{3} - 1 = -\frac{4}{3} - \frac{3}{3} = -\frac{7}{3} \approx -2.33 \][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] are:
[tex]\[ \left( 3.33, -2.33 \right) \][/tex]
Given points [tex]\( A(x_1, y_1) = (2, -1) \)[/tex] and [tex]\( B(x_2, y_2) = (4, -3) \)[/tex], let's calculate the coordinates step-by-step:
1. We know that [tex]\( P \)[/tex] divides the segment [tex]\( AB \)[/tex] in the ratio [tex]\( \frac{2}{3} \)[/tex]. Therefore, we can denote [tex]\( m = 2 \)[/tex] and [tex]\( n = 1 \)[/tex]. This comes from the ratio [tex]\( 2 \)[/tex] parts of [tex]\( AB \)[/tex] to [tex]\( 1 \)[/tex] part of [tex]\( AP \)[/tex] added together giving [tex]\( m + n = 3 \)[/tex].
2. Substitute the ratio and coordinates into the section formula:
[tex]\[ x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1 \][/tex]
[tex]\[ y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1 \][/tex]
Plugging in the values:
[tex]\[ x = \left(\frac{2}{3}\right)(4 - 2) + 2 \][/tex]
Now calculate the [tex]\( x \)[/tex]-coordinate:
[tex]\[ x = \left(\frac{2}{3}\right)(2) + 2 = \frac{4}{3} + 2 = \frac{4}{3} + \frac{6}{3} = \frac{10}{3} \approx 3.33 \][/tex]
Now for the [tex]\( y \)[/tex]-coordinate:
[tex]\[ y = \left(\frac{2}{3}\right)(-3 - (-1)) + (-1) \][/tex]
[tex]\[ y = \left(\frac{2}{3}\right)(-2) - 1 = -\frac{4}{3} - 1 = -\frac{4}{3} - \frac{3}{3} = -\frac{7}{3} \approx -2.33 \][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] are:
[tex]\[ \left( 3.33, -2.33 \right) \][/tex]
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.