Use the image to answer the question.

An illustration shows a coordinate plane with the x axis extending from negative 11 to 2 and the y axis extending from negative 3 to 10 in increments of 1. Points upper A,upper B, and upper C are plotted in quadrant 2. Upper A is plotted at left parenthesis negative 9 comma 4 right parenthesis. Upper B is plotted at left parenthesis negative 11 comma 1 right parenthesis. Upper C is plotted at left parenthesis negative 2 comma 1 right parenthesis.

Point A
and Point D
have the same second coordinate, and ABCD
is a quadrilateral. Line segment AD
is 3 cm in length. What coordinates does Point D
need to be plotted at to make this statement true?

(1 point)
Responses

(−2, 4)
left parenthesis negative 2 comma 4 right parenthesis

(4, −6)
left parenthesis 4 comma negative 6 right parenthesis

(−6, 4)
left parenthesis negative 6 comma 4 right parenthesis

(6, −4)
left parenthesis 6 comma negative 4 right parenthesis

1 answer

To find the coordinates of Point D, we know that Point D must have the same second coordinate (y-coordinate) as Point A, which is 4. Therefore, we can represent Point D as (x, 4).

Additionally, we are given that line segment AD is 3 cm in length. To calculate the distance between points A and D, we can use the distance formula in the context of the coordinate plane.

Point A is at (-9, 4), so we have:

  • The y-coordinate of Point D is 4.
  • The x-coordinate of Point D can be determined by the distance formula:

\[ \text{Distance} = |x_D - x_A| = |x_D - (-9)| = |x_D + 9| \] We need this distance to equal 3 cm. Therefore,

\[ |x_D + 9| = 3 \]

This equation gives us two possible scenarios:

  1. \(x_D + 9 = 3\) which leads to \(x_D = 3 - 9 = -6\)
  2. \(x_D + 9 = -3\) which leads to \(x_D = -3 - 9 = -12\)

Since Point D must be plotted on the coordinate plane, we need to ensure that the x-coordinate is within a reasonable range based on the provided information.

Clearly, the only suitable answer based on the choices given is \((-6, 4)\).

Thus, the coordinates of Point D are: \[ \text{(−6, 4)} \]