Question

the endpoints of a line segment are (-6, 13) and (11, 5). which of the following is the midpoint and the length of the segment?
(\frac{5}{2}, 4), \sqrt{353} units
(\frac{5}{2}, 9), \sqrt{353} units
(\frac{5}{2}, 4), \sqrt{89} units
(\frac{5}{2}, 9), \sqrt{89} units

Explanation:

Step1: Find the mid - point

The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Given \((x_1,y_1)=(-6,13)\) and \((x_2,y_2)=(11,5)\). Then \(\frac{-6 + 11}{2}=\frac{5}{2}\) and \(\frac{13+5}{2}=\frac{18}{2} = 9\). So the mid - point is \((\frac{5}{2},9)\).

Step2: Find the length

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Substitute \(x_1=-6,y_1 = 13,x_2=11,y_2 = 5\) into the formula: \((x_2 - x_1)=11-(-6)=17\), \((y_2 - y_1)=5 - 13=-8\). Then \(d=\sqrt{17^2+(-8)^2}=\sqrt{289 + 64}=\sqrt{353}\) units.

Answer:

B. \((\frac{5}{2},9),\sqrt{353}\) units