Answer:
The y-intercept of AB is y=4/3, the x-coordinate of the poit C is 4, the equation of the line that goes through BC is ![6x-11=y](https://tex.z-dn.net/?f=6x-11%3Dy)
Step-by-step explanation:
To find the y-intercept of AB we first have to find the equation of the line that goes through this segment.
We firt compute the slope of the segement which is given by
![m_1 = \dfrac{y_B - y_A}{x_B - x_A}= \dfrac{1-(-1)}{2-14} =\dfrac{-1}{6}](https://tex.z-dn.net/?f=m_1%20%3D%20%5Cdfrac%7By_B%20-%20y_A%7D%7Bx_B%20-%20x_A%7D%3D%20%5Cdfrac%7B1-%28-1%29%7D%7B2-14%7D%20%3D%5Cdfrac%7B-1%7D%7B6%7D)
Using this we can compute the equation of the line through AB using the point B=(2,1) and the formula
![\dfrac{-1}{6}=\dfrac{1-y}{2-x}](https://tex.z-dn.net/?f=%5Cdfrac%7B-1%7D%7B6%7D%3D%5Cdfrac%7B1-y%7D%7B2-x%7D)
we solve the last equation for
and so we get
![y=\dfrac{-x}{6}+\dfrac{4}{3}](https://tex.z-dn.net/?f=y%3D%5Cdfrac%7B-x%7D%7B6%7D%2B%5Cdfrac%7B4%7D%7B3%7D)
The y-intercept of AB is the point in the line with x-coordinate equals to 0. Hence, the y-intercept is
![y(0) = \frac{-0}{6}+\frac{4}{3}=\frac{4}{3}](https://tex.z-dn.net/?f=y%280%29%20%3D%20%5Cfrac%7B-0%7D%7B6%7D%2B%5Cfrac%7B4%7D%7B3%7D%3D%5Cfrac%7B4%7D%7B3%7D)
Now, since the segments AB and BC are perpendicular, the slope of the line that goes through the segment BC is
![m_2 = -\dfrac{1}{m_1}=-\dfrac{1}{-\frac{1}{6}}=6](https://tex.z-dn.net/?f=m_2%20%3D%20-%5Cdfrac%7B1%7D%7Bm_1%7D%3D-%5Cdfrac%7B1%7D%7B-%5Cfrac%7B1%7D%7B6%7D%7D%3D6)
Usinge the slope
and knowing that the line goes through the point B=(2,1), to find the equation of the line through the segmen AC we use the following formula
![6=\dfrac{y-1}{x-2}](https://tex.z-dn.net/?f=6%3D%5Cdfrac%7By-1%7D%7Bx-2%7D)
we solve the equation for y and we get the equation
![y=6x-11](https://tex.z-dn.net/?f=y%3D6x-11)
Finally, to compute the x-coordinate of the point C we use the last equation and the fact that the y-coordinate is 13, it holds that
![13 = 6 x - 11 \quad \Rightarrow \quad x = \dfrac{13 + 11}{6}=\dfrac{24}{6}=4](https://tex.z-dn.net/?f=13%20%3D%206%20x%20-%2011%20%5Cquad%20%5CRightarrow%20%5Cquad%20x%20%3D%20%5Cdfrac%7B13%20%2B%2011%7D%7B6%7D%3D%5Cdfrac%7B24%7D%7B6%7D%3D4)