Question

Find the length of line segment ab

Answer 1

the intersect when both graphs are equal to each other, so we can say

`\begin{cases} y=-3x+7\\\\ y=x^2-4x+1 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ -3x+7~~ = ~~x^2-4x+1\implies 7=x^2-x+1\implies 0=x^2-x-6 \\\\\\ 0=(x-3)(x+2) \implies x= \begin{cases} 3\\ -2 \end{cases}`

now, when "x" is those values, for the line, let's find out what's "y".

`y(3)=-3(3)+7\implies y(3)=-2\hspace{7em}(3~~,~-2) \\\\\\ y(-2)=-3(-2)+7\implies y(-2)=13\hspace{5em}(-2~~,~~13)`

now, let's use those two points at the intersection of the two graphs to get their distance

`~~~~~~~~~~~~\textit{distance between 2 points} \\\\ A(\stackrel{x_1}{3}~,~\stackrel{y_1}{-2})\qquad B(\stackrel{x_2}{-2}~,~\stackrel{y_2}{13})\qquad \qquad d = √(( x_2- x_1)^2 + ( y_2- y_1)^2) \\\\\\ AB=√((~~-2 - 3~~)^2 + (~~13 - (-2)~~)^2)\implies AB=√((-2 -3)^2 + (13 +2)^2) \\\\\\ AB=√( (-5)^2 + (15)^2) \implies AB=√( 25 + 225) \\\\\\ AB=√( 250 )\implies AB\approx 15.81`