Question

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

A. (8,6)

B. (5,11)

C. (-1,19)

D. (11,3)​

Answer 1

d

Explanation:

Answer 2

D. (11,3)​

Explanation:

First of all, let's plot each point as indicated in figure below. To find the point
`P(x,y)` that divide a segment with endpoints
`P_(1)(x_(1),y_(1)) \ and \ P_(2)(x_(2),y_(2))` and a ratio:

`r=\frac{\left | \overline{P_(1)P}\right |}{\left | \overline{PP_(2)} \right |}`

We must use the formula:

`(x,y)=\left((x_(1)+rx_(2))/(1+r),(y_(1)+ry_(2))/(1+r)\right)`

To find the ratio, we know that:

`\left | \overline{P_(1)P}\right |=(1)/(6)`

because
`P` is 1/6 of the way from
`A(14,-1)` to
`B(-4,23)`

So the other part of the segment is:

`\left | \overline{PP_(2)} \right |=(5)/(6)`

Therefore, the ratio can be found as:

`r=\frac{\left | \overline{P_(1)P}\right |}{\left | \overline{PP_(2)} \right |}=((1)/(6))/((5)/(6))=(1)/(5)`

From here, we can calculate the point we are looking for:

`(x,y)=\left((14+(1/5)(-4))/(1+1/5),(-1+(1/5)(23))/(1+1/5)\right) \\ \\ (x,y)=(11,3) \\ \\ \\ Finally: \\ \\ \boxed{P(x,y)=P(11,3)}`