Question

for the points (5, 4) and (2, 1), (a) find the exact distance between the points. (b) find the midpoint of the line segment whose endpoints are the given points.

Explanation:

Step1: Recall distance formula

The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $x_1 = 5,y_1 = 4,x_2=2,y_2 = 1$.

Step2: Substitute values into formula

$d=\sqrt{(2 - 5)^2+(1 - 4)^2}=\sqrt{(-3)^2+(-3)^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}$

Step3: Recall mid - point formula

The mid - point $M$ of the line segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$ is $M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.

Step4: Substitute values into mid - point formula

$M=(\frac{5+2}{2},\frac{4 + 1}{2})=(\frac{7}{2},\frac{5}{2})$

Answer:

(a) $3\sqrt{2}$
(b) $(\frac{7}{2},\frac{5}{2})$