Line through two points, distance and midpoint

Text

Given the points $A(1,2)$ and $B(5,10)$.

• (a) Write the equation of the line $AB$.
• (b) Compute the length of the segment $AB$ and the coordinates of its midpoint.

Solution

(a) The slope is $m=\dfrac{10-2}{5-1}=2$. The line through $A$ is $y-2=2(x-1)$, i.e. $y=2x$. (b) Distance: $|AB|=\sqrt{(5-1)^{2}+(10-2)^{2}}=\sqrt{16+64}=\sqrt{80}=4\sqrt5\approx 8.944.$ Midpoint: $M\!\left(\dfrac{1+5}{2},\dfrac{2+10}{2}\right)=M(3,6)$. $\boxed{\,y=2x,\quad |AB|=4\sqrt5\approx 8.944,\quad M(3,6)\,}$