Respuesta :
Answer:
The coordinates of point C are (-8,5).
Step-by-step explanation:
It is given that A, B and C collinear and B is between A and C.
The ratio of AB to BC is 1:2. It means Point divided the line segments AC in 1:2.
Section formula:
[tex](\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n})[/tex]
The given points are A(7,-1) and B(2,1).
Let the coordinates of C are (a,b).
Using section formula the coordinates of B are
[tex]B=(\dfrac{(1)(a)+(2)(7)}{1+2},\dfrac{(1)(b)+(2)(-1)}{1+2})[/tex]
[tex]B=(\dfrac{a+14}{3},\dfrac{b-2}{3})[/tex]
We know that point B(2,1).
[tex](2,1)=(\dfrac{a+14}{3},\dfrac{b-2}{3})[/tex]
On comparing both sides we get
[tex]2=\dfrac{a+14}{3}[/tex]
[tex]6=a+14[/tex]
[tex]6-14=a[/tex]
[tex]-8=a[/tex]
The value of a is -8.
[tex]1=\dfrac{b-2}{3}[/tex]
[tex]3=b-2[/tex]
[tex]3+2=b[/tex]
[tex]5=b[/tex]
The value of b is 5.
Therefore, the coordinates of point C are (-8,5).
The coordinates of the point C such that points A and B are (7, -1) and (2, 1) and the ratio AB to BC is 1 : 2 is (-8, 5).
How to determine the location of a point within a line segment
According to the Euclidean geometry, a line is formed by two points on a plane and three points are collinear if all the three points go through a single line.
By definitions of vector and ratio we derive an expression to determine the coordinates of the point B:
[tex]\overrightarrow{AB} = \frac{1}{1+2}\cdot \overrightarrow{AC}[/tex]
[tex]\vec B - \vec A = \frac{1}{3}\cdot \vec C -\frac{1}{3}\cdot \vec A[/tex]
[tex]\frac{1}{3}\cdot \vec C = \vec B - \frac{2}{3}\cdot \vec A[/tex]
[tex]\vec C = 3 \cdot \vec B - 2\cdot \vec A[/tex]
If we know that A(x,y) = (7, -1) and B(x,y) = (2, 1), then the coordinates of point C is:
C(x, y) = 3 · (2, 1) - 2 · (7, -1)
C(x, y) = (6, 3) + (- 14, 2)
C(x,y) = (- 8, 5)
The coordinates of the point C such that points A and B are (7, -1) and (2, 1) and the ratio AB to BC is 1 : 2 is (-8, 5).
Remark
The statement is poorly formatted and reports mistakes. Correct form is shown below:
A, B and C are collinear and B is between A and C. The ratio of AB to BC is 1 : 2. If A is A(x, y) = (7, -1) and B(x, y) = (2, 1), what are the coordinates of point C?
To learn more on line segments, we kindly invite to check this verified question: https://brainly.com/question/25727583
