Question

the coordinates of the endpoints of $overline{qr}$ are $q(8,2)$ and $r(5,7)$. which measurement is closest to the length of $overline{qr}$ in units? a 5 units b 3.9 units c 4 units d 5.8 units

Explanation:

Step1: Identify distance - formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $Q(8,2)$ means $x_1 = 8,y_1 = 2$ and $R(5,7)$ means $x_2 = 5,y_2 = 7$.

Step2: Substitute values

Substitute the values into the formula:
\[

$$\begin{align*} d&=\sqrt{(5 - 8)^2+(7 - 2)^2}\\ &=\sqrt{(- 3)^2+5^2}\\ &=\sqrt{9 + 25}\\ &=\sqrt{34} \end{align*}$$

\]

Step3: Approximate the value

We know that $\sqrt{25}=5$ and $\sqrt{36}=6$. Since $34$ is closer to $36$ than $25$, $\sqrt{34}\approx5.8$.

Answer:

D. 5.8 units