Find the distance between the points left parenthesis negative 1 comma 2 right parenthesis and left parenthesis 3 comma 4 right parenthesis. Express your answer in simplest radical form.

Question

(1 point)
Responses

2 Start Root 5 End Root
Image with alt text: 2 Start Root 5 End Root

Start Root 8 End Root
Image with alt text: Start Root 8 End Root

Start Root 20 End Root
Image with alt text: Start Root 20 End Root

2 Start Root 2 End Root
Image with alt text: 2 Start Root 2 End Root

Answer 1

To find the distance between the points $(-1,2)$ and $(3,4)$, we can use the distance formula:

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$$

Here, $(x_1,y_1)=(-1,2)$ and $(x_2,y_2)=(3,4)$.

Now, we can plug in the values:

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

Calculating the differences:

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

Now, square the differences:

$$d=\sqrt{16+4}=\sqrt{20}$$

Now, simplify $\sqrt{20}$:

$$\sqrt{20}=\sqrt{4\cdot 5}=2\sqrt{5}$$

Thus, the distance between the points $(-1,2)$ and $(3,4)$ is:

$$\boxed{{\displaystyle 2\sqrt{5}}}$$