Question

for the point ( p(-22,-1) ) and ( q(-15,4) ), find the distance ( d(p,q) ) and the coordinates of the midpoint ( m ) of the segment ( pq ).
what is the distance?
(simplify your answer. type an exact answer, using radicals as need.)
what are the coordinates of the midpoint ( m )?
(simplify your answer. type an ordered pair, using integers or fractions.)

Explanation:

Step1: Calculate the distance between two points

The distance formula between two points \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Here, \(x_1=-22,y_1 = - 1,x_2=-15,y_2 = 4\).
\(x_2-x_1=-15-(-22)=-15 + 22 = 7\), \(y_2-y_1=4-(-1)=4 + 1=5\).
\(d=\sqrt{7^2+5^2}=\sqrt{49 + 25}=\sqrt{74}\)

Step2: Calculate the mid - point of the segment

The mid - point formula for two points \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) is \(M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
\(\frac{x_1+x_2}{2}=\frac{-22-15}{2}=\frac{-37}{2}\), \(\frac{y_1 + y_2}{2}=\frac{-1 + 4}{2}=\frac{3}{2}\)

Answer:

The distance \(d=\sqrt{74}\).
The coordinates of the mid - point \(M\) is \((-\frac{37}{2},\frac{3}{2})\).