Question

Given 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.

a) (2, 1)
b) (0, 1)
c) (-2, 1)
d) (-1, 1)

Answer 1

The coordinates of point Y, which divides line segment XZ one-fifth of the way from X to Z, are (-2, 1) (option c).

Step-by-step explanation:

To find the coordinates of point Y, which divides line segment XZ one-fifth of the way from X to Z, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint (Y) between two points (X and Z) can be found by taking the average of the x-coordinates and the average of the y-coordinates. In this case, the x-coordinate of Y can be found by taking (-4 + (1/5)(6 - (-4)) = -2, and the y-coordinate of Y can be found by taking

(3 + (1/5)(-2 - 3)) = 1.

Therefore, the coordinates of Y are (-2, 1), which corresponds to option c) in the given choices.

Answer 2

The coordinates of Y if Y divides XZ one-fifth of the way from X to Z is (-2, 1)

Therefore, correct answer is c) (-2, 1)

Step-by-step explanation:

To find the coordinates of Y, which divides XZ one-fifth of the way from X to Z, we use the section formula. Let Y be the point \((x, y)\). The coordinates of Y can be found using the formula:

`\[ x = \frac{{(1 - k) \cdot x_1 + k \cdot x_2}}{{1 + k}}, \quad y = \frac{{(1 - k) \cdot y_1 + k \cdot y_2}}{{1 + k}} \]`

Here,
`\(k = (1)/(5)\)`, X(-4, 3), and Z(6, -2). Plugging in these values, we find Y(-2, 1).

The section formula is a powerful tool to find the coordinates of a point that divides a line segment into a given ratio. It's essential to understand the formula and how to apply it, especially when dealing with geometric problems involving points and segments.

Therefore, correct answer is c) (-2, 1)