Question

1. what is the distance between points a(-1,4) and b(2,0)

a. $sqrt{32}$
b. $sqrt{64}$
c. 0
d. $sqrt{32}$

Explanation:

Step1: Identify distance - formula

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

Step2: Substitute the coordinates

Here, $x_1=-1,y_1 = 4,x_2 = 2,y_2=0$. So $d=\sqrt{(2-(-1))^2+(0 - 4)^2}$.

Step3: Simplify the expression

First, calculate $(2-(-1))^2+(0 - 4)^2=(2 + 1)^2+(-4)^2=9 + 16=25$. Then $d=\sqrt{25}=5$. But if we calculate step - by - step in terms of the formula form:
\[

$$\begin{align*} d&=\sqrt{(2-(-1))^2+(0 - 4)^2}\\ &=\sqrt{(3)^2+(-4)^2}\\ &=\sqrt{9 + 16}\\ &=\sqrt{25}\\ & = 5 \end{align*}$$

\]
We can also rewrite $\sqrt{32}=\sqrt{16\times2}=4\sqrt{2}$, $\sqrt{64} = 8$. Since the correct distance calculation gives $\sqrt{(2+1)^2+(0 - 4)^2}=\sqrt{9 + 16}=\sqrt{25}=5$, and $\sqrt{32}
eq5,\sqrt{64}
eq5,0
eq5$. There is an error in the options provided. If we calculate strictly by the formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=\sqrt{(2+1)^2+(0 - 4)^2}=\sqrt{9+16}=\sqrt{25}$.

Answer:

None of the options A. $\sqrt{32}$, B. $\sqrt{64}$, C. $0$, D. $\sqrt{32}$ are correct. The correct distance is $5$.