Answer:
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
