Good morning. I have another math problem I just need checked a bit please until I learn how to check and get proficient at checking with the distance formula.
#9
Given segment XZ with X(-4, 3) and Z(6, -2), find the coordinates of Y if Y divides XZ one-fifth of the way from X to Z.
-4 + 1/5 (6 - -4)
-4 + 1/5 (10)
-4 + 10/5
-4 + 2= -2
X=-2
3 + 1/5 (-2 -3)
3 + 1/5 (-5)
3 + -5/5
3 + -1 = 2
Y=2
Are my answers correct?
change in x = 10/5 =2 so x = -2
change in y = -5/5 = -1 so y = 2
So I agree
Hello Damon is this the easiest way to double check the answers by doing change in x and change in y or do you have to use the distance formula to check your answers?
You do not need the distance formula.
in general
1/5 of distance from (x1,y1) to (x2,y2)
D = sqrt [(x2-x1)^2 + (y2-y1)^2 ]
we want d = D/5
if xf = (x2-x1)/5 and yf = (y2-y1)/5
d = sqrt [ (x2-x1)^2/ 25 + (y2-y1)^2 / 25 ]
d = 1/5 sqrt [(x2-x1)^2 + (y2-y1)^2 ] sure enough
To check if your answers for the coordinates of point Y are correct, you can use the distance formula. The distance formula is √[(x₂ - x₁)² + (y₂ - y₁)²].
Given that X(-4, 3), Y(-2, 2), and Z(6, -2), we can calculate the distances:
Distance XZ = √[(6 - -4)² + (-2 - 3)²]
= √(10² + (-5)²)
= √(100 + 25)
= √125
= 5√5
Distance XY = √[(-2 - -4)² + (2 - 3)²]
= √(2² + (-1)²)
= √(4 + 1)
= √5
Distance YZ = √[(6 - -2)² + (-2 - 2)²]
= √(8² + (-4)²)
= √(64 + 16)
= √80
= 4√5
Now, we can check if the ratio of the distances XY:YZ is 1:4 (one-fifth):
XY:YZ = (Distance XY / Distance XZ):(Distance YZ / Distance XZ)
= (√5 / 5√5):(4√5 / 5√5)
= 1:4
The ratio is indeed 1:4, which confirms that Y divides XZ one-fifth of the way from X to Z. Therefore, your answers for the coordinates of Y are correct.