A, b, and c are collinear, and B is between a and c. The ratio of AB to AC is 1:2. If A is at (7,-1) and B is at (2,1) what are
the coordinates of point C
1 answer:
Answer:
C(-3,3)
Step-by-step explanation:
Given
A = (7,-1)
B = (2,1)
AB:AC = 1:2
Required
Determine the coordinates of C
Since, B is between A and C; we need to determine ratio BC as follows;
![AB:AC = 1:2](https://tex.z-dn.net/?f=AB%3AAC%20%3D%201%3A2)
Convert to division
![\frac{AB}{AC} = \frac{1}{2}](https://tex.z-dn.net/?f=%5Cfrac%7BAB%7D%7BAC%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D)
<em>AC = AB + BC;</em>
![\frac{AB}{AB + BC} = \frac{1}{2}](https://tex.z-dn.net/?f=%5Cfrac%7BAB%7D%7BAB%20%2B%20BC%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D)
Cross Multiply
![2 * AB = 1 * (AB + BC)](https://tex.z-dn.net/?f=2%20%2A%20AB%20%3D%201%20%2A%20%28AB%20%2B%20BC%29)
![2 AB = AB + BC](https://tex.z-dn.net/?f=2%20AB%20%3D%20AB%20%2B%20BC)
![2AB - AB = BC](https://tex.z-dn.net/?f=2AB%20-%20AB%20%3D%20BC)
![AB = BC](https://tex.z-dn.net/?f=AB%20%3D%20BC)
Divide both sides by BC
![\frac{AB}{BC} = 1](https://tex.z-dn.net/?f=%5Cfrac%7BAB%7D%7BBC%7D%20%3D%201)
Rewrite as
![\frac{AB}{BC} = \frac{1}{1}](https://tex.z-dn.net/?f=%5Cfrac%7BAB%7D%7BBC%7D%20%3D%20%5Cfrac%7B1%7D%7B1%7D)
Write as ratio
![AB:BC = 1:1](https://tex.z-dn.net/?f=AB%3ABC%20%3D%201%3A1)
Next is to determine the coordinates of C as follows;
Because B is between both points. we have:
![B(x,y) = (\frac{mx_2 + nx_1}{m+n},\frac{my_2 + ny_1}{m+n})](https://tex.z-dn.net/?f=B%28x%2Cy%29%20%3D%20%28%5Cfrac%7Bmx_2%20%2B%20nx_1%7D%7Bm%2Bn%7D%2C%5Cfrac%7Bmy_2%20%2B%20ny_1%7D%7Bm%2Bn%7D%29)
Where
![m:n = AB:BC = 1:1](https://tex.z-dn.net/?f=m%3An%20%3D%20AB%3ABC%20%3D%201%3A1)
![B(x,y) = B(2,1)](https://tex.z-dn.net/?f=B%28x%2Cy%29%20%3D%20B%282%2C1%29)
![A(x_1,y_1) = A(7,-1)](https://tex.z-dn.net/?f=A%28x_1%2Cy_1%29%20%3D%20A%287%2C-1%29)
So; we're solving for x2 and y2
![B(2,1) = (\frac{mx_2 + nx_1}{m+n},\frac{my_2 + ny_1}{m+n})](https://tex.z-dn.net/?f=B%282%2C1%29%20%3D%20%28%5Cfrac%7Bmx_2%20%2B%20nx_1%7D%7Bm%2Bn%7D%2C%5Cfrac%7Bmy_2%20%2B%20ny_1%7D%7Bm%2Bn%7D%29)
Where
Solving for x2;
![x = \frac{mx_2 + nx_1}{m+n}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7Bmx_2%20%2B%20nx_1%7D%7Bm%2Bn%7D)
![2 = \frac{1 * x_2 + 1 * 7}{1+1}](https://tex.z-dn.net/?f=2%20%3D%20%5Cfrac%7B1%20%2A%20x_2%20%2B%201%20%2A%207%7D%7B1%2B1%7D)
![2 = \frac{x_2 + 7}{2}](https://tex.z-dn.net/?f=2%20%3D%20%5Cfrac%7Bx_2%20%2B%20%207%7D%7B2%7D)
Cross Multiply
![2 * 2 = x_2 + 7](https://tex.z-dn.net/?f=2%20%2A%202%20%3D%20x_2%20%2B%207)
![4 = x_2 + 7](https://tex.z-dn.net/?f=4%20%3D%20x_2%20%2B%207)
![x_2 = 4 - 7](https://tex.z-dn.net/?f=x_2%20%3D%204%20-%207)
![x_2 = -3](https://tex.z-dn.net/?f=x_2%20%3D%20-3)
Solving for y2;
![y = \frac{my_2 + ny_1}{m+n}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7Bmy_2%20%2B%20ny_1%7D%7Bm%2Bn%7D)
![1 = \frac{1 * y_2 + 1 * -1}{1+1}](https://tex.z-dn.net/?f=1%20%3D%20%5Cfrac%7B1%20%2A%20y_2%20%2B%201%20%2A%20-1%7D%7B1%2B1%7D)
![1 = \frac{y_2- 1}{2}](https://tex.z-dn.net/?f=1%20%3D%20%5Cfrac%7By_2-%201%7D%7B2%7D)
Cross Multiply
![2 * 1 = y_2 - 1](https://tex.z-dn.net/?f=2%20%2A%201%20%3D%20y_2%20-%201)
![2 = y_2 - 1](https://tex.z-dn.net/?f=2%20%20%3D%20y_2%20-%201)
![y_2 = 2 + 1](https://tex.z-dn.net/?f=y_2%20%20%3D%202%20%2B%201)
![y_2 = 3](https://tex.z-dn.net/?f=y_2%20%20%3D%203)
<em>Hence, the coordinates of C are: C(-3,3)</em>
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