SOLUTION: Asking for help Pls... The line segment with endpoints A(-1,-6) and B(3,0) Is extended beyond point A to point C so that C is 4 times as far from B as from A. Find the coordinates

Algebra ->  Length-and-distance -> SOLUTION: Asking for help Pls... The line segment with endpoints A(-1,-6) and B(3,0) Is extended beyond point A to point C so that C is 4 times as far from B as from A. Find the coordinates       Log On


   

Question 691362: Asking for help Pls... The line segment with endpoints A(-1,-6) and B(3,0) Is extended beyond point A to point C so that C is 4 times as far from B as from A. Find the coordinates of point C.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Since C is a point on an extension of segment AB it must lie on the same line as segment AB. So let's start by finding the equation of the line which contains segment AB.

m%5BAB%5D+=+%280-%28-6%29%29%2F%283-%28-1%29%29+=+6%2F4+=+3%2F2
From the point-slope form:
y+-+0+=+3%2F2%28x-3%29
which simplifies to:
y+=+%283%2F2%29x-9%2F2
Point C must be on this line which means its coordinates must fit this equation.

Next let's use the distance formula to find the length of segment AB:


We want the length of segment BC to be 4 times as much. So
BC+=+4%2A%282sqrt%2813%29%29+=+8sqrt%2813%29

If C is the point (x%5BC%5D, y%5BC%5D) then we can use the distance formula to express the length of segment BC:
BC+=+sqrt%28%28x%5BC%5D-3%29%5E2+%2B+%28y%5BC%5D-0%29%5E2%29

The two expressions we have for BC must be equal to each other:
8sqrt%2813%29+=+sqrt%28%28x%5BC%5D-3%29%5E2+%2B+%28y%5BC%5D-0%29%5E2%29
(Note: The order of the following steps can be changed.) I'm going to start by eliminating the square roots by squaring both sides:
%288sqrt%2813%29%29%5E2+=+%28sqrt%28%28x%5BC%5D-3%29%5E2+%2B+%28y%5BC%5D-0%29%5E2%29%29%5E2
which simplifies to:
832+=+%28x%5BC%5D-3%29%5E2+%2B+%28y%5BC%5D-0%29%5E2%29
Continuing to simplify:
832+=+x%5BC%5D%5E2-6x%2B9+%2B+y%5BC%5D%5E2
Since point C is on the line containing segment AB it coordinates must fit the equation we found:
y%5BC%5D+=+%283%2F2%29x%5BC%5D-9%2F2
Substituting for y%5BC%5D into the previous equation:
832+=+x%5BC%5D%5E2-6x%2B9+%2B+%28%283%2F2%29x%5BC%5D-9%2F2%29%5E2
Simplifying:
832+=+x%5BC%5D%5E2-6x%2B9+%2B+%289%2F4%29x%5BC%5D%5E2-%2854%2F4%29x%5BC%5D+%2B+81%2F4
Multiplying both sides by 4 will eliminate the fractions:
3328+=+4x%5BC%5D%5E2-24x%2B36+%2B+9x%5BC%5D%5E2-54x%5BC%5D+%2B+81
Combining like terms:
3328+=+13x%5BC%5D%5E2-78x%5BC%5D%2B117

Now we solve for x%5BC%5D. Subtracting 3328:
0+=+13x%5BC%5D%5E2-78x%5BC%5D%2B3211
Now we factor. First the GCF of 13:
0+=+13%28x%5BC%5D%5E2-13x%5BC%5D%2B247%29
Next, believe it or not, the trinomial factors:
0+=+13%28x%5BC%5D-19%29%28x%5BC%5D%2B13%29
From the Zero Product Property:
x%5BC%5D-19+=+0 or x%5BC%5D%2B13+=+0
Solving these we get:
x%5BC%5D=19 or x%5BC%5D+=+-13

It should not surprise us that we have two answers. There should be two points on the line that are 8sqrt%2813%29 away from B, one in each direction. The point we are interested in is in the same direction from B as A is. As we go from B to A the x coordinate changes from 3 to -1. If we continue in the same direction the x coordinates should continue to go down. So the x coordinate of C is -13, not 19.

Now we can find C's y coordinate. Using the equation of the line:
y%5BC%5D+=+%283%2F2%29x%5BC%5D-9%2F2
and replacing the x%5BC%5D with -13:
y%5BC%5D+=+%283%2F2%29%28-13%29-9%2F2
Simplifying...
y%5BC%5D+=+%28-39%29%2F2-9%2F2
y%5BC%5D+=+%28-48%29%2F2
y%5BC%5D+=+-24
So C is the point (-13, -24)