Question

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

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

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

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

(−6, 4)
left parenthesis negative 6 comma 4 right parenthesis
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Answer 1

To find the coordinates of Point D such that it has the same second coordinate as Point A and lies 3 cm away from Point A, we start with the coordinates of Point A, which are \((-9, 4)\).

Since Point D must have the same y-coordinate as Point A, its y-coordinate should also be \(4\). We can represent Point D as \((x, 4)\).

Now, we need to find an x-coordinate for Point D such that the distance between Point A \((-9, 4)\) and Point D \((x, 4)\) is \(3\) cm.

The distance formula in this case is simplified because the y-coordinates are the same:

\[
\text{Distance} = |x_2 - x_1| = |x - (-9)| = |x + 9|
\]

We want this distance to be equal to 3:

\[
|x + 9| = 3
\]

This absolute value equation gives us two scenarios:

1. \(x + 9 = 3\)
- Solving this gives:
\[
x = 3 - 9 = -6
\]

2. \(x + 9 = -3\)
- Solving this gives:
\[
x = -3 - 9 = -12
\]

Thus, the possible coordinates for Point D are:

- At \((-6, 4)\) (which is valid as it lies within the bounds on the coordinate plane)
- At \((-12, 4)\) (which is outside the range of x-values on the provided coordinate plane)

Now, looking at the answer choices provided:

- (4, −6)
- (6, −4)
- (−2, 4)
- (−6, 4)

The only valid coordinate for Point D that meets the criteria of having the same y-coordinate as Point A and being 3 cm away is **\((-6, 4)\)**.

Thus, the answer is:

\[
\text{Point D coordinates: } (-6, 4)
\]