On a coordinate plane, a line is drawn from point k to point j. point k is at (9, 2) and point j is at (1, negative 10). what is the x-coordinate of the point that divides the directed line segment from k to j into a ratio of 1: 3? x = (startfraction m over m + n endfraction) (x 2 minus x 1) + x 1 –1 3 7 11

the x coordinate is 7 if the shorter segment goes first, and the x coordinate is 3 if the shorter segment goes at the end ( after the longer one)

Step-by-step explanation:

Besides the line that goes through K (9,2) to J (1,-10) , we can define a point as L (1,2) such that it makes a equilateral triangle KJL.

- the side LJ (a vertical line in x=1, from y=2 until from y=-10  ) has a length= 2 - (-10) = 12

- the side KL (an horizontal line in y=2, from x=9 until from x=1  ) has a length= 9 - 1 =8

- the line KJ has a length = (9-1)²+ [2-(-10)]² = 8² + 12² =  64 + 144 = 208

now we can define another point P over the line KJ such that it divides the segment in a ratio of 1:3 --> one part of the segment has 3 times the lenght of the other -->

that means the shorter segment with lenght = 208 *1 /(3+1) =  208/4 = 52

and the longer one = 208 * 3/(3+1) = 156

Now here we have two options

1) KP is the segment of length 52 and PJ has 156

2) PJ is the segment of length 52 and KP has 156

in case 1) we can define another point P2 over KL ( the horizontal line) that has the x coordinate that we want such that

KPP2 is another equilateral triangle inside of the triangle KJL that shares the K corner --> since both are have the same angle in K we can say

sin of the angle formed between sides KJ and KL = sin of the angle formed between sides KP and KP2

that is

KJ/KL= KP/KP2

208/8=52/KP2

KP2= 52* 8/208 = 2

- the side KP2 (the horizontalline in y=2, from x=9 until from x=X ) has a length= 9 - X

KP2 = 9 - X

X = 9 - KP2 = 9 - 2 = 7

the x- coordinate is 7

in case 2) is the same reasoning but KP =156

KJ/KL= KP/KP2

208/8=156/KP2

KP2= 156 * 8/208 = 6

- the side KP2 (a vertical line in y=2, from x=9 until from x=X ) has a length= 9 - X

KP2 = 9 - X

X = 9 - KP2 = 9 - 6 = 3

the x- coordinate is 3

what is all that apply?