The distance YZ is 3.8-1.3 = 2.5
Y is 5/12 of the way from X to Z.
So, YZ is 7/5 as big as XY
So, Z = 3.8 + (7/5)(2.5) = 7.3
Y is 5/12 of the way from X to Z.
So, YZ is 7/5 as big as XY
So, Z = 3.8 + (7/5)(2.5) = 7.3
Given that the ratio of the length of XY segment to the length of YZ segment is 5:7, we can set up the following equation:
\( \frac{XY}{YZ} = \frac{5}{7} \)
Let's denote the length of XY as \( a \) and the length of YZ as \( b \). Using this notation, we can rewrite the equation as:
\( \frac{a}{b} = \frac{5}{7} \)
We can also determine the length of XY and YZ by calculating the difference in coordinates:
\( XY = Y - X = 3.8 - 1.3 = 2.5 \)
So, we now have:
\( \frac{2.5}{b} = \frac{5}{7} \)
To solve for \( b \), we can cross multiply:
\( 2.5 \times 7 = 5 \times b \)
\( 17.5 = 5b \)
Finally, divide both sides of the equation by 5 to solve for \( b \):
\( b = \frac{17.5}{5} \)
Evaluating this expression, we find that:
\( b = 3.5 \)
Now, we can determine the length of YZ:
\( YZ = b = 3.5 \)
To find the coordinate of Z, we add the length of YZ to the coordinate of Y:
\( Z = Y + YZ = 3.8 + 3.5 \)
Evaluating this expression, we find that:
\( Z = 7.3 \)
Therefore, the coordinate of point Z is 7.3.
Let's first find the length of XY segment. We are given that the ratio of XY to YZ is 5:7. Since the total ratio is 5 + 7 = 12, we can determine that the length of XY is (5/12) times the total length of XZ.
Next, we need to find the length of XZ. The coordinate of X is given as 1.3, and the coordinate of Y is 3.8. Therefore, the length of XZ can be calculated as the absolute difference between the coordinates of X and Y, which is |3.8 - 1.3|.
Now, let's substitute the values into the equation to find the length of XY:
Length of XY = (5/12) * Length of XZ
Substituting the values, we get:
Length of XY = (5/12) * |3.8 - 1.3|
Calculating the absolute difference inside the bracket, we get:
Length of XY = (5/12) * |2.5|
Length of XY = (5/12) * 2.5
Length of XY = 1.04
Now that we know the length of XY, we can find the length of YZ by subtracting the length of XY from the length of XZ:
Length of YZ = Length of XZ - Length of XY
Length of YZ = |3.8 - 1.3| - 1.04
Length of YZ = 2.5 - 1.04
Length of YZ = 1.46
Finally, to find the coordinate of Z, we need to add the length of YZ to the coordinate of Y:
Coordinate of Z = Coordinate of Y + Length of YZ
Coordinate of Z = 3.8 + 1.46
Coordinate of Z = 5.26
Therefore, the coordinate of Z is 5.26.