Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Get immediate and reliable answers to your questions from a community of experienced experts on our platform. Get quick and reliable solutions to your questions from a community of experienced experts on our 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] into the ratio 5:8, follow these steps:
1. Identify the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- Point [tex]\( A \)[/tex] has coordinates [tex]\( (x_1, y_1) = (-22, -6.3) \)[/tex].
- Point [tex]\( B \)[/tex] has coordinates [tex]\( (x_2, y_2) = (-2.4, -6.4) \)[/tex].
2. Identify the ratio [tex]\( m:n \)[/tex]:
- The ratio [tex]\( m:n = 5:8 \)[/tex], where [tex]\( m = 5 \)[/tex] and [tex]\( n = 8 \)[/tex].
3. Calculate the [tex]\( x \)[/tex]-coordinate of point [tex]\( C \)[/tex]:
[tex]\[ x = \left( \frac{m}{m+n} \right) \left( x_2 - x_1 \right) + x_1 \][/tex]
Substituting the values:
[tex]\[ x = \left( \frac{5}{5+8} \right) \left( -2.4 - (-22) \right) + (-22) \][/tex]
[tex]\[ x = \left( \frac{5}{13} \right) \left( -2.4 + 22 \right) - 22 \][/tex]
[tex]\[ x = \left( \frac{5}{13} \right) \left( 19.6 \right) - 22 \][/tex]
[tex]\[ x = \left( 0.384615 \right) \left( 19.6 \right) - 22 \][/tex]
[tex]\[ x = 7.5384 - 22 \][/tex]
[tex]\[ x \approx -14.5 \][/tex]
4. Calculate the [tex]\( y \)[/tex]-coordinate of point [tex]\( C \)[/tex]:
[tex]\[ y = \left( \frac{m}{m+n} \right) \left( y_2 - y_1 \right) + y_1 \][/tex]
Substituting the values:
[tex]\[ y = \left( \frac{5}{5+8} \right) \left( -6.4 - (-6.3) \right) + (-6.3) \][/tex]
[tex]\[ y = \left( \frac{5}{13} \right) \left( -6.4 + 6.3 \right) + (-6.3) \][/tex]
[tex]\[ y = \left( \frac{5}{13} \right) \left( -0.1 \right) + (-6.3) \][/tex]
[tex]\[ y = \left( 0.384615 \right) \left( -0.1 \right) + (-6.3) \][/tex]
[tex]\[ y = -0.0385 + (-6.3) \][/tex]
[tex]\[ y \approx -6.3 \][/tex]
5. State the coordinates of point [tex]\( C \)[/tex]:
- The [tex]\( x \)[/tex]-coordinate of point [tex]\( C \)[/tex] is approximately [tex]\( -14.5 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate of point [tex]\( C \)[/tex] is approximately [tex]\( -6.3 \)[/tex].
Thus, the coordinates of point [tex]\( C \)[/tex], which partitions the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] into the ratio 5:8, are [tex]\( (-14.5, -6.3) \)[/tex] rounded to the nearest tenth.
1. Identify the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- Point [tex]\( A \)[/tex] has coordinates [tex]\( (x_1, y_1) = (-22, -6.3) \)[/tex].
- Point [tex]\( B \)[/tex] has coordinates [tex]\( (x_2, y_2) = (-2.4, -6.4) \)[/tex].
2. Identify the ratio [tex]\( m:n \)[/tex]:
- The ratio [tex]\( m:n = 5:8 \)[/tex], where [tex]\( m = 5 \)[/tex] and [tex]\( n = 8 \)[/tex].
3. Calculate the [tex]\( x \)[/tex]-coordinate of point [tex]\( C \)[/tex]:
[tex]\[ x = \left( \frac{m}{m+n} \right) \left( x_2 - x_1 \right) + x_1 \][/tex]
Substituting the values:
[tex]\[ x = \left( \frac{5}{5+8} \right) \left( -2.4 - (-22) \right) + (-22) \][/tex]
[tex]\[ x = \left( \frac{5}{13} \right) \left( -2.4 + 22 \right) - 22 \][/tex]
[tex]\[ x = \left( \frac{5}{13} \right) \left( 19.6 \right) - 22 \][/tex]
[tex]\[ x = \left( 0.384615 \right) \left( 19.6 \right) - 22 \][/tex]
[tex]\[ x = 7.5384 - 22 \][/tex]
[tex]\[ x \approx -14.5 \][/tex]
4. Calculate the [tex]\( y \)[/tex]-coordinate of point [tex]\( C \)[/tex]:
[tex]\[ y = \left( \frac{m}{m+n} \right) \left( y_2 - y_1 \right) + y_1 \][/tex]
Substituting the values:
[tex]\[ y = \left( \frac{5}{5+8} \right) \left( -6.4 - (-6.3) \right) + (-6.3) \][/tex]
[tex]\[ y = \left( \frac{5}{13} \right) \left( -6.4 + 6.3 \right) + (-6.3) \][/tex]
[tex]\[ y = \left( \frac{5}{13} \right) \left( -0.1 \right) + (-6.3) \][/tex]
[tex]\[ y = \left( 0.384615 \right) \left( -0.1 \right) + (-6.3) \][/tex]
[tex]\[ y = -0.0385 + (-6.3) \][/tex]
[tex]\[ y \approx -6.3 \][/tex]
5. State the coordinates of point [tex]\( C \)[/tex]:
- The [tex]\( x \)[/tex]-coordinate of point [tex]\( C \)[/tex] is approximately [tex]\( -14.5 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate of point [tex]\( C \)[/tex] is approximately [tex]\( -6.3 \)[/tex].
Thus, the coordinates of point [tex]\( C \)[/tex], which partitions the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] into the ratio 5:8, are [tex]\( (-14.5, -6.3) \)[/tex] rounded to the nearest tenth.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
