Question

type the correct answer in each box.use numerals instead of words. if necessary, use / for the fraction bar.
\\( \overleftrightarrow { a b } \\) and \\( \overleftrightarrow { b c } \\) form a right angle at their point of intersection, b.
if the coordinates of a and b are (14, -1) and (2, 1), respectively, the y - intercept of \\( \overleftrightarrow { a b } \\) is and the equation of \\( \overleftrightarrow { b c } \\)
is \\( y = \\) x +.
if the y - coordinate of point c is 13, its x - coordinate is

Explanation:

Step1: Find the equation of \(\overleftrightarrow{AB}\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(A(14,-1)\) and \(B(2,1)\), \(m_{AB}=\frac{1-(-1)}{2 - 14}=\frac{2}{-12}=-\frac{1}{6}\).
Using the point - slope form \(y - y_1=m(x - x_1)\) with point \(B(2,1)\): \(y - 1=-\frac{1}{6}(x - 2)\).
Simplify to slope - intercept form \(y=-\frac{1}{6}x+\frac{1}{3}+1=-\frac{1}{6}x+\frac{4}{3}\). The \(y\) - intercept is \(\frac{4}{3}\).

Step2: Find the slope of \(\overleftrightarrow{BC}\)

Since \(\overleftrightarrow{AB}\perp\overleftrightarrow{BC}\), if \(m_{AB}=-\frac{1}{6}\), then \(m_{BC}=6\) (because the product of slopes of perpendicular lines is \(- 1\), \(m_{AB}\times m_{BC}=-1\)).
Using the point - slope form with point \(B(2,1)\) and \(m = 6\): \(y - 1=6(x - 2)\).
Simplify to \(y=6x-12 + 1=6x-11\).

Step3: Find the \(x\) - coordinate of point \(C\)

If \(y = 13\) in the equation \(y=6x-11\), then \(13=6x-11\).
Add \(11\) to both sides: \(13 + 11=6x\), \(24=6x\).
Divide both sides by \(6\): \(x = 4\).

Answer:

The \(y\) - intercept of \(\overleftrightarrow{AB}\) is \(\frac{4}{3}\), the equation of \(\overleftrightarrow{BC}\) is \(y = 6x-11\), and the \(x\) - coordinate of point \(C\) is \(4\).