Question

rs has an endpoint at r(6, - 4) and length 17. which of the following cannot be the coordinates of s? choose the correct answer below. a. (23,13) b. (6,13) c. (-9, - 12) d. (23, - 4) e. (14,11)

Explanation:

Step1: Recall distance formula

The distance formula between two points $R(x_1,y_1)$ and $S(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here $x_1 = 6,y_1=-4$ and $d = 17$.

Step2: Check option A

For $S(23,13)$:
\[

$$\begin{align*} d&=\sqrt{(23 - 6)^2+(13+ 4)^2}\\ &=\sqrt{17^2+17^2}\\ &=\sqrt{2\times17^2}\\ &=17\sqrt{2} eq17 \end{align*}$$

\]

Step3: Check option B

For $S(6,13)$:
\[

$$\begin{align*} d&=\sqrt{(6 - 6)^2+(13 + 4)^2}\\ &=\sqrt{0+17^2}\\ &=17 \end{align*}$$

\]

Step4: Check option C

For $S(-9,-12)$:
\[

$$\begin{align*} d&=\sqrt{(-9 - 6)^2+(-12 + 4)^2}\\ &=\sqrt{(-15)^2+(-8)^2}\\ &=\sqrt{225 + 64}\\ &=\sqrt{289}\\ &=17 \end{align*}$$

\]

Step5: Check option D

For $S(23,-4)$:
\[

$$\begin{align*} d&=\sqrt{(23 - 6)^2+(-4 + 4)^2}\\ &=\sqrt{17^2+0}\\ &=17 \end{align*}$$

\]

Step6: Check option E

For $S(14,11)$:
\[

$$\begin{align*} d&=\sqrt{(14 - 6)^2+(11 + 4)^2}\\ &=\sqrt{8^2+15^2}\\ &=\sqrt{64+225}\\ &=\sqrt{289}\\ &=17 \end{align*}$$

\]

Answer:

A. $(23,13)$