Question

For the points (√5, √2) and (4√5, -5√2).(a) Find the exact distance between the points.(b) Find the midpoint of the line segment whose endpoints are the given points

Answer 1

(a)3√13 Units

(b)

`((5√(5))/(2),-2√(2))`

Step-by-step explanation:

Part A

We determine the distance between the two points using the distance formula.

`\text{Distance}=√((x_2-x_1)^2+(y_2-y_1)^2)`

Therefore, the distance between points (√5, √2) and (4√5, -5√2) is:

`\begin{gathered} =\sqrt{(4√(5)-√(5))^2+(-5√(2)-√(2))^2} \\ =\sqrt{(3√(5))^2+(-6√(2))^2} \\ =√((3^2*5)+((-6)^2*2)) \\ =√(45+72) \\ =√(117) \\ =3√(13)\text{ Units} \end{gathered}`

The exact distance between the points is 3√13 Units.

Part B

We determine the midpoint of the line segment whose endpoints are the given points using the midpoint formula.

`\text{Midpoint}=((x_1+x_2)/(2),(y_1+y_2)/(2))`

Therefore:

`\begin{gathered} \text{Midpoint}=((√(5)+4√(5))/(2),(√(2)+(-5√(2)))/(2)) \\ =((5√(5))/(2),(-4√(2)))/(2)) \\ =((5√(5))/(2),-2√(2)) \end{gathered}`