Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Discover a wealth of knowledge from experts across different disciplines on our comprehensive Q&A platform. 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]\( C \)[/tex], which partitions the directed line segment from point [tex]\( A \)[/tex] to point [tex]\( B \)[/tex] in the ratio [tex]\( 5:8 \)[/tex], we will use the section formula.
Given:
- Coordinates of [tex]\( A \)[/tex] are [tex]\( (-2.2, -6.3) \)[/tex]
- Coordinates of [tex]\( B \)[/tex] are [tex]\( (2.7, -0.7) \)[/tex]
- Ratio [tex]\( m:n = 5:8 \)[/tex]
The section formula for the coordinates of a point [tex]\( C \)[/tex] that divides the line segment joining [tex]\( A (x_1, y_1) \)[/tex] and [tex]\( B (x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[ \begin{align*} x &= \left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1, \\ y &= \left(\frac{m}{m+n}\right) \left(y_2 - y_1\right) + y_1. \end{align*} \][/tex]
First, let's find the [tex]\( x \)[/tex]-coordinate of [tex]\( C \)[/tex]:
[tex]\[ x = \left(\frac{5}{5+8}\right) \left(2.7 - (-2.2)\right) + (-2.2). \][/tex]
Simplify within the parentheses:
[tex]\[ x = \left(\frac{5}{13}\right) \left(2.7 + 2.2\right) - 2.2. \][/tex]
Calculate the sum:
[tex]\[ x = \left(\frac{5}{13}\right) \times 4.9 - 2.2. \][/tex]
Divide [tex]\( 4.9 \)[/tex] by [tex]\( 13 \)[/tex] and then multiply by [tex]\( 5 \)[/tex]:
[tex]\[ x = \left(\frac{5 \times 4.9}{13}\right) - 2.2 = \frac{24.5}{13} - 2.2 = 1.8846 - 2.2. \][/tex]
Subtract [tex]\( 2.2 \)[/tex]:
[tex]\[ x = -0.3. \][/tex]
Next, let's find the [tex]\( y \)[/tex]-coordinate of [tex]\( C \)[/tex]:
[tex]\[ y = \left(\frac{5}{5+8}\right) \left(-0.7 - (-6.3)\right) + (-6.3). \][/tex]
Simplify within the parentheses:
[tex]\[ y = \left(\frac{5}{13}\right) \left(-0.7 + 6.3\right) - 6.3. \][/tex]
Calculate the sum:
[tex]\[ y = \left(\frac{5}{13}\right) \times 5.6 - 6.3. \][/tex]
Divide [tex]\( 5.6 \)[/tex] by [tex]\( 13 \)[/tex] and then multiply by [tex]\( 5 \)[/tex]:
[tex]\[ y = \left(\frac{5 \times 5.6}{13}\right) - 6.3 = \frac{28}{13} - 6.3 = 2.1538 - 6.3. \][/tex]
Subtract [tex]\( 6.3 \)[/tex]:
[tex]\[ y = -4.1. \][/tex]
Therefore, the coordinates of point [tex]\( C \)[/tex] are [tex]\( (-0.3, -4.1) \)[/tex], rounded to the nearest tenth.
Given:
- Coordinates of [tex]\( A \)[/tex] are [tex]\( (-2.2, -6.3) \)[/tex]
- Coordinates of [tex]\( B \)[/tex] are [tex]\( (2.7, -0.7) \)[/tex]
- Ratio [tex]\( m:n = 5:8 \)[/tex]
The section formula for the coordinates of a point [tex]\( C \)[/tex] that divides the line segment joining [tex]\( A (x_1, y_1) \)[/tex] and [tex]\( B (x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[ \begin{align*} x &= \left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1, \\ y &= \left(\frac{m}{m+n}\right) \left(y_2 - y_1\right) + y_1. \end{align*} \][/tex]
First, let's find the [tex]\( x \)[/tex]-coordinate of [tex]\( C \)[/tex]:
[tex]\[ x = \left(\frac{5}{5+8}\right) \left(2.7 - (-2.2)\right) + (-2.2). \][/tex]
Simplify within the parentheses:
[tex]\[ x = \left(\frac{5}{13}\right) \left(2.7 + 2.2\right) - 2.2. \][/tex]
Calculate the sum:
[tex]\[ x = \left(\frac{5}{13}\right) \times 4.9 - 2.2. \][/tex]
Divide [tex]\( 4.9 \)[/tex] by [tex]\( 13 \)[/tex] and then multiply by [tex]\( 5 \)[/tex]:
[tex]\[ x = \left(\frac{5 \times 4.9}{13}\right) - 2.2 = \frac{24.5}{13} - 2.2 = 1.8846 - 2.2. \][/tex]
Subtract [tex]\( 2.2 \)[/tex]:
[tex]\[ x = -0.3. \][/tex]
Next, let's find the [tex]\( y \)[/tex]-coordinate of [tex]\( C \)[/tex]:
[tex]\[ y = \left(\frac{5}{5+8}\right) \left(-0.7 - (-6.3)\right) + (-6.3). \][/tex]
Simplify within the parentheses:
[tex]\[ y = \left(\frac{5}{13}\right) \left(-0.7 + 6.3\right) - 6.3. \][/tex]
Calculate the sum:
[tex]\[ y = \left(\frac{5}{13}\right) \times 5.6 - 6.3. \][/tex]
Divide [tex]\( 5.6 \)[/tex] by [tex]\( 13 \)[/tex] and then multiply by [tex]\( 5 \)[/tex]:
[tex]\[ y = \left(\frac{5 \times 5.6}{13}\right) - 6.3 = \frac{28}{13} - 6.3 = 2.1538 - 6.3. \][/tex]
Subtract [tex]\( 6.3 \)[/tex]:
[tex]\[ y = -4.1. \][/tex]
Therefore, the coordinates of point [tex]\( C \)[/tex] are [tex]\( (-0.3, -4.1) \)[/tex], rounded to the nearest tenth.
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.