Question

Segment RS has endpoints at R(5,-3) and S(5,13) , Point Q lies on RS such that RQ : QS = 3 : 1 . Which of the following must be on the coordinates of Q?
(1) (5,1)
(2) (17,3)
(3) (5,9)
(4) (6,4)

Answer 1

Answer:

(3) Q(5,9)

Step-by-step explanation:

Let point Q has coordinates (x,y). Consider vectors $\overrightarrow{RQ}$ and $\overrightarrow{QS}.$ Note that

• $\overrightarrow{RQ}=(x-5,y+3);$
• $\overrightarrow{QS}=(5-x,13-y).$

Since $\dfrac{RQ}{QS}=\dfrac{3}{1},$ then

$\dfrac{\overrightarrow{RQ}}{\overrightarrow{QS}}=\dfrac{3}{1},$

Thus,

$\dfrac{x-5}{5-x}=\dfrac{3}{1}\Rightarrow x-5=3(5-x),\ x-5=15-3x,\ 4x=20,\ x=5.\\ \\\dfrac{y+3}{13-y}=\dfrac{3}{1}\Rightarrow y+3=3(13-y),\ y+3=39-3y,\ 4y=36,\ y=9.$