Answer:
Step-by-step explanation:
Segments AB and BC intersect each other at 90° at B.
Let the equation of the segment AB → y = mx + b
Here, m = Slope of the line
b = y-intercept
Slope of the line AB passing through A(14, -1) and B(2, 1)
Slope = ![\frac{y_2-y_1}{x_2-x_1}](https://tex.z-dn.net/?f=%5Cfrac%7By_2-y_1%7D%7Bx_2-x_1%7D)
= ![\frac{1+1}{2-14}](https://tex.z-dn.net/?f=%5Cfrac%7B1%2B1%7D%7B2-14%7D)
= ![-\frac{2}{12}](https://tex.z-dn.net/?f=-%5Cfrac%7B2%7D%7B12%7D)
= ![-\frac{1}{6}](https://tex.z-dn.net/?f=-%5Cfrac%7B1%7D%7B6%7D)
Equation of the line will be,
![y=-\frac{1}{6}(x)+b](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B1%7D%7B6%7D%28x%29%2Bb)
Since, AB passes through (2, 1)
![1=-\frac{1}{6}(2)+b](https://tex.z-dn.net/?f=1%3D-%5Cfrac%7B1%7D%7B6%7D%282%29%2Bb)
![b=1+\frac{1}{3}](https://tex.z-dn.net/?f=b%3D1%2B%5Cfrac%7B1%7D%7B3%7D)
![b=\frac{4}{3}](https://tex.z-dn.net/?f=b%3D%5Cfrac%7B4%7D%7B3%7D)
Therefore, y-intercept of AB = ![\frac{4}{3}](https://tex.z-dn.net/?f=%5Cfrac%7B4%7D%7B3%7D)
Equation of AB → ![y=-\frac{1}{6}(x)+\frac{4}{3}](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B1%7D%7B6%7D%28x%29%2B%5Cfrac%7B4%7D%7B3%7D)
Since, line BC is perpendicular to AB,
By the property of perpendicular lines,
![m_1\times m_2=-1](https://tex.z-dn.net/?f=m_1%5Ctimes%20m_2%3D-1)
Here,
and
are the slopes of line AB and BC respectively.
By this property,
![-\frac{1}{6}\times m_2=-1](https://tex.z-dn.net/?f=-%5Cfrac%7B1%7D%7B6%7D%5Ctimes%20m_2%3D-1)
![m_2=6](https://tex.z-dn.net/?f=m_2%3D6)
Equation of a line passing through a point (h, k) and slope 'm' is,
(y - k) = m(x - h)
Therefore, equation of line BC passing through B(2, 1) and slope = 6,
y - 1 = 6(x - 2)
y = 6x - 11
Since, line BC passes through C(x, 13),
13 = 6x - 11
6x = 24
x = 4
Therefore, x-coordinate of point C will be, x = 4