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Answer:
y-intercept of AB is 4/3.
The equation of line BC is y=6x-11.
The x-coordinate of point C is 4.
Step-by-step explanation:
Given information: AB⊥BC, A(14,-1) and B(2,1).
If a line passes through two points then the equation of line is
[tex]y-y_1=m(x-x_1)[/tex]
where, m is slope.
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
Slope of line AB is
[tex]m_{AB}=\frac{1-(-1)}{2-14}=-\frac{1}{6}[/tex]
The equation of line AB is
[tex]y-(-1)=-\frac{1}{6}(x-14)[/tex]
[tex]y+1=-\frac{1}{6}(x)+\frac{14}{6}[/tex]
[tex]y=-\frac{1}{6}(x)+\frac{7}{3}-1[/tex]
[tex]y=-\frac{1}{6}(x)+\frac{4}{3}[/tex]
The equation of line AB is [tex]y=-\frac{1}{6}(x)+\frac{4}{3}[/tex].
Substitute x=0 to find the y-intercept of AB.
[tex]y=-\frac{1}{6}(0)+\frac{4}{3}[/tex]
[tex]y=\frac{4}{3}[/tex]
Therefore y-intercept of AB is 4/3.
Line AB and BC are perpendicular, so product of their slopes is -1.
[tex]m_{BC}=-\frac{1}{m_{AB}}=6[/tex]
Slope of BC is 6 and it passes through (2,1). So equation of BC is
[tex]y-(1)=6(x-2)[/tex]
[tex]y-1=6x-12[/tex]
[tex]y=6x-12+1[/tex]
[tex]y=6x-11[/tex]
Therefore the equation of line BC is y=6x-11.
Point C lies on line BC. The y-coordinate of C is 13.
Substitute y=13 in the above equation.
[tex]13=6x-11[/tex]
Add 11 both sides.
[tex]13+11=6x[/tex]
[tex]24=6x[/tex]
Divide both sides by 6.
[tex]4=x[/tex]
Therefore the x-coordinate of point C is 4.
The y-intercept of AB is 4/3.
The equation of line BC is y = 6x-11.
The x-coordinate of point C is 4.
Given
AB and BC form a right angle at their point of intersection, B.
If the coordinates of A and Bare (14, -1) and (2, 1), respectively.
What is a linear equation?
An equation between two variables that gives a straight line when plotted on a graph.
[tex]\rm y=mx+c[/tex]
Where; m = Slope of the line, c = y-intercept.
The slope of the line AB passing through A(14, -1) and B(2, 1) is;
[tex]\rm Slope = \dfrac{1+1}{2-14}\\\\Slope = \dfrac{2}{-12}\\\\Slope=\dfrac{1}{-6}\\[/tex]
Then,
The equation of line will be;
[tex]\rm y=mx+c\\\\1=\dfrac{-1}{6}(2)+b\\\\1+\dfrac{1}{3}=b\\\\b= \dfrac{1+3}{3}\\\\b=\dfrac{4}{3}[/tex]
The y-intercept of AB is 4/3.
The line BC is perpendicular to AB,
By the property of perpendicular lines,
[tex]\rm m_1\times m_2=-1\\\\\dfrac{-1}{6}\times m_2=-1\\\\ m_2=6[/tex]
Equation of a line passing through a point (h, k) and slope 'm' is,
(y - k) = m(x - h)
Therefore, the equation of line BC passing through B(2, 1) and slope = 6,
[tex]\rm y - 1 = 6(x - 2)\\\\y = 6x - 11[/tex]
The equation of line BC is y = 6x-11.
Since, line BC passes through C(x, 13),
[tex]\rm 13 = 6x - 11\\\\6x = 24\\\\x = 4[/tex]
The x-coordinate of point C is 4.
To know more about the Linear equations click the link given below.
https://brainly.com/question/13138556
