What are the coordinates for the point on Line AB that is

The point needs to be 3 times farther from one point than the other,

this implies, the line can be divided in the ratio 1:3 or 4 parts.

This would mean you need to find 2 mid points.

A(-5,6) and B (-2,0).

First midpoint M1 ((-5-2)/2, (6-0)/2 ) = M1 (-3.5,3)

Second mid point needs to be in between B and M1 and hence M2 ((-3.5-2)/2 , (3-0)/2) = M2(-2.75,1.5)

Yes, you don't need any formulas.

Note that from B to A, x co-ordinate decreases by 3 units and y co-ordinate increases by 6 units.
You need to divide this distance equally into 4 units (3 units away from A and 1 unit away from B).
For each unit, x co-ordinate will decrease by 3/4 and y co-ordinate will increase by 6/4
Since we need 1 unit away from B, x co-ordinate of th required point becomes -2 - 3/4 = -2.75 and y co-ordinate becomes 0 + 6/4 = 1.5

Answer (-2.75, 1.5)

Solution:

First we can find M, the midpoint of A and B:

M = ((-5 + (-2))/2, (6 + 0)/2) = (-7/2, 3)

Next, we find N, the midpoint of M and B:

N = (-7/2 + (-2))/2, (3 + 0)/2) = (-11/4, 3/2)

Notice that now N is three times as far from A as from B.

Answer: (-11/4, 3/2)
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Hi ScottTargetTestPrep,

First you figured out M (Midpoint) which has to be equidistant from A and B. Then you figured out N (Midpoint of M and B). Then how is N three times as far from A as from B.

Three times as far from A as from B means it has to be 3 times as far from both A as well as B right?

It'll be great if you'd help me understand

Thank you

Posted from my mobile device

Hi Narenn,

How did you figure out m=3 and n=1?

Secondly, doesn't "Three times as far from A as from B" mean that the point has to be 3 times far from both A as well as B?

Please help me understand if I'm missing out on something.

Thank you.

Posted from my mobile device

Three times as far from A as from B means the distance between the point and A is equal to three times the distance between the point and B. So, if the distance between N and B is d, then the distance between N and A should be 3d so that we can say the point N is three times as far from A as from B.
To elaborate further, M is the midpoint of A and B, so the distance between A and M is equal to the distance between M and B. Let's call this distance s. N is the midpoint between M and B, so the distance between M and N is equal to the distance between N and B. Since we already represented the distance between M and B by s, the distance between M and N is s/2 and the distance between N and B is also s/2.
Notice that the distance between A and N is the sum of the distance between A and M and the distance between M and N. The former is s and the latter is s/2, so the distance between A and N is s + s/2 = 3s/2, which is precisely three times the distance between N and B, which is s/2.
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The simplest way to solve this question is by using the midpoint formula. We need to find a point that is 3times as far from A as B. So consider the midpoint of A and B, Let's call this point C((-5+-2)/2; (6+0)/2). Now the midpoint of point C and B will be exactly 3 times as far from A as B. So the midpoint of C & B let's call it D will have the co-ordinates((-3.5-2/2;3-0/2).

No answer choices in the question! Please edit

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Done. Thank you.
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