You can think of this in the following way.Anantjit wrote:What are the coordinates for the point on line AB that is three times as far as from A as from B, and that is in between points A and B.Point A is (-5.6) Point B is (-2,0)
Points A and B are at the corners of a right triangle, the hypotenuse of which is segment AB and the third vertex of which is point (-5,0).
So the length of the vertical side is 6 - 0 = 6 and of the horizontal side is -5 - (-2) = 3.
We need to find a point 1/4 of the way from B to A.
We can use similar triangles to come up with the coordinates of that point. The lengths of all three sides of similar triangles are in the same ratio with the lengths of their corresponding sides.
So if one side of a similar triangle is 1/4 of the length of the corresponding side of the other triangle, then the lengths of the other two sides will also be 1/4 of the lengths of the corresponding sides in the other triangle.
So we are going to create a triangle similar to the triangle that we have and with all sides 1/4 of the lengths of the sides of the triangle we have.
1/4 of 6 is 1.5. So the length of the vertical side of our smaller similar triangle is 1.5.
Starting at B, go up 1.5, to get a y coordinate of 1.5
1/4 of 3 is .75. So the length of the horizontal side of our similar triangle is .75.
Starting at B go .75 to the left, to get an x coordinate of -2.75.
So the the three vertices of our new triangle are B, which is (-2,O), (-2.75,0) and the point we are looking for, which is (-2.75,1.5).
The correct answer is [spoiler](-2.75,1.5)[/spoiler].
