Question

Find the coordinates of the missing endpoint if W is the MIDPOINT of XY.

a) (6,13)
b) (4,13)
c) (6,5)
d) (4,5)

Answer 1

The coordinates of the missing endpoint, calculated using the midpoint formula, are (6, 13), corresponding to option a) in the answer choices. The correct option is a) (6,13).

Step-by-step explanation:

To find the coordinates of the missing endpoint, we can use the midpoint formula, which states that the coordinates of the midpoint (W) between two points (X and Y) can be found by averaging their respective x-coordinates and y-coordinates. The formula is:

`\[ W = \left((X_x + Y_x)/(2), (X_y + Y_y)/(2)\right) \]`

Given that W is the midpoint of XY, and the coordinates of W (6, 9) are provided, we can set up two equations based on the formula for the x-coordinate and the y-coordinate:

`\[ 6 = (X_x + Y_x)/(2) \]`

`\[ 9 = (X_y + Y_y)/(2) \]`

Solving for the missing coordinates, we can multiply both sides of each equation by 2:

`\[ X_x + Y_x = 12 \]`

`\[ X_y + Y_y = 18 \]`

Now, we know that X_x + Y_x is the sum of the x-coordinates, and X_y + Y_y is the sum of the y-coordinates. Since W is the midpoint, it has the coordinates (6, 9). Subtracting W's coordinates from the sum gives us the missing endpoint's coordinates:

`\[ (X_x, X_y) = (12 - 6, 18 - 9) = (6, 9) \]`

Therefore, the missing endpoint is (6, 9), which corresponds to option a) in the answer choices.

Answer 2

The missing endpoint of XY, given that W is the midpoint, is (4,5) as per option d).

Step-by-step explanation:

The midpoint formula for finding the coordinates of the midpoint given two endpoints (X and Y) is
`\(_((x_m,y_m)) = \left(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}\right)\)`. Here, we are given that W is the midpoint of XY, which implies that the coordinates of W,
`\(_((x_m,y_m))\)`, are equal to the average of the coordinates of X and Y.

Given W is at (4,5), and assuming X's coordinates are (x, y), we can use the midpoint formula to find Y's coordinates. As W is the midpoint, the sum of the x-coordinates of X and Y divided by 2 equals 4, and the sum of their y-coordinates divided by 2 equals 5.

Let's represent X's coordinates as (x, y). Plugging in the values and solving for Y's coordinates:

`- \(_{\left(\frac{{x + x}}{2}, \frac{{y + y}}{2}\right)} = (4,5)\)`

`- \(_{\left(\frac{{2x}}{2}, \frac{{2y}}{2}\right)} = (4,5)\)`

`- \(_((x, y)) = (4,5)\)`

Therefore, the missing endpoint is (4,5), aligning with option d).