Question

Points A (-10,-6) and B (6,2) are the endpoints of AB. What are the coordinates of point C on AB such that AC is 3/4 the length of AB?
a. (0,-1)
b. (2,0)
c. (-2,-2)
d. (4,1)

Answer 1

@Owlcoffee

Answer 2

Are you fond to vectors?

Answer 3

Thanks

Answer 4

No

Answer 5

Actually yes I am.

Answer 6

Okay, so I'll show you the vector way.
We can express the vector whose tail is located on "A" and head on "B" like this:
\[\vec a ((6-(-10)), (2-(-6))) \rightarrow \vec a (16,8)\]
\(\vec a\) is a vector that goes from A to B, and we want to find a "C" on AB such that AC=3/4 the length of AB.
This means a 3/4 part of the vector, this creates a new vector \(\vec w\) that is \(\frac{ 3 }{ 4 } \vec a\), but we can do it by expressing the vector on it's referential form:
\[\vec a = 16 \vec i + 8 \vec j \rightarrow \frac{ 3 }{ 4 } \vec a = (\frac{ 3 }{ 4 } 16) \vec i + (\frac{ 3 }{ 4 }8) \vec j \]
\[\frac{ 3 }{ 4 } \vec a = \vec w \]
\[\vec w = (3.4) \vec i + (3.2)\vec j\]
\[\vec w = 12 \vec i + 6 \vec j\]
This implies that the new coordinates lie on \[\vec w (12,6)\]
And this implies, regressing the operation of the vector, since the tail still lies on A, but the head is now on the point C:
\[x_c-(-10)=12\]
\[y_c - (-6)=6\]

Answer 7

Did it kick everyone offline?

Answer 8

No, just some changes in the website.

Answer 9

So it would be D?

Answer 10

no, solve the first equation for xc and the other for yc, that'll give you the coordinates of the point C

Answer 11

So B

Answer 12

correct.

Answer 13

When I said I was fond of vectors I ment in science...

Answer 14

Gave you a medal... Thanks so much.

Answer 15

Oh.. Well, if I were to say the method wihout them, I would suggest you find the distance between those two points.
Then multiply it by 3/4, and then the process is completely analogous.