Points X and Y are on a number line, and Y partitions XZ segment into two parts so that the of the length of XY segment to the length of YZ segment is 5:7. The coordinate of x is 1.3, and the coordinate of Y is 3.8. What is the coordinate of Z?
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
sorry; XY=2.5, not YZ
To find the coordinate of point Z, we need to determine the length of the XY and YZ segments.
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.
To find the coordinate of Z, we can use the concept of proportionality.
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.