The x- and y- coordinates of point P on the directed line segment from A to B is \(\left({3,-2}\right)\)
Further explanation:
Given:
The options of the point P are as follows.
a.\(\left( {2,1}\right)\)
b.\(\left({4,-3}\right)\)
c.\(\left({-1,2}\right)\)
d.\(\left( {3,-2}\right)\)
Explanation:
The x-coordinate and y- coordinate of point P on the directed line segment from A to B such that P is \(\dfrac{2}{3}\) the length of the line segment from A to B.
The given ratio is \(\dfrac{2}{3}.\)
The line intersect the points that are \(\left( {-6,7}\right)\) and \(\left( {9,-8}\right).\)
The slope of the line can be obtained as follows.
\(\begin{aligned}m&=\frac{{ - 8 - 7}}{{9 + 6}}\\&=\frac{{ - 15}}{{15}}\\&= - 1\\\end{aligned}\)
They-coordinate point P can obtain as follows,
\(\begin{aligned}y- {\text{coordinate}}&= \frac{2}{3}\left( {{y_2} - {y_1}} \right)\\&= \frac{2}{3}\left( { - 15}\right)\\&= - 10\\\end{aligned}\)
Now start from point A and move point 10 units in upward direction.
The x-coordinate point P can obtain as follows,
\(\begin{aligned}x - {\text{coordinate}}&=\frac{2}{3}\left( {{x_2} - {x_1}} \right)\\&=\frac{2}{3}\left( {15}\right)\\&= 10\\\end{aligned}\)
Now start from point A and move point 10 units in left direction.
The x- and y- coordinates of point P on the directed line segment from A to B is \(\left({ - 1,2}\right).\)
Option (a) is not correct as the coordinates of point P is \(\left( { - 1,2}\right).\)
Option (b) is not correct as the coordinates of point P is \(\left( { - 1,2}\right).\)
Option (c) is correct as the coordinates of point P is \(\left({ - 1,2}\right).\)
Option (d) is not correct as the coordinates of point P is \(\left( { - 1,2}\right).\)
Hence, \(\boxed{{\text{Option (c)}}}\) is correct.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Linear inequalities
Keywords: numbers, slope, slope intercept, inequality, equation, linear equation, shaded region, y-intercept, graph, representation, origin, perpendicular, x-intercept 6.
