Question

What are the coordinates of the point that is 1/6 of the way from A(14, −1) to B(−4, 23)?

Answer 1

Answer:  The correct option is

(A) (11, 3).

Step-by-step explanation:  We are given to find the co-ordinates of he point that is one-sixth of the way from A(14, −1) to B(−4, 23).

As shown in the attached figure below, let point P is one-sixth of the way from A to B.

Also, let P divides the the segment AB in the ratio m : n.

Then, we must have

$\dfrac{m}{m+n}=\dfrac{1}{6}\\\\\\\Rightarrow 6m=m+n\\\\\Rightarrow 6m-m=n\\\\\Rightarrow 5m=n\\\\\Rightarrow \dfrac{m}{n}=\dfrac{1}{5}\\\\\Rightarrow m:n=1:5.$

So, the point P divides AB in the ratio 1 : 5.

We know that

if a point divides the line segment joining the points (a, b) and (c, d) in the ratio m : n, then its co-ordinates are

$\left(\dfrac{mc+na}{m+n},\dfrac{md+nb}{m+n}\right).$

Therefore, the co-ordinates of the point P will be

$\left(\dfrac{1\times(-4)+5\times14}{1+5},\dfrac{1\times23+5\times(-1)}{1+5}\right)\\\\\\=\left(\dfrac{-4+70}{6},\dfrac{23-5}{6}\right)\\\\\\=\left(\dfrac{66}{6},\dfrac{18}{6}\right)\\\\=(11,3).$

Thus, the required co-ordinates of the point is (11, 3).

Option (A) is CORRECT.

Answer 2

What are the coordinates of the point that is 1/6 of the way from A(14, −1) to B(−4, 23)?
A) (11, 3)