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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

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

(−2, 4)

A (-9,4)
B(-11,1)
C(-2,1)

1 answer

To determine the coordinates for Point D given that it must have the same second coordinate as Point A and that the length of line segment AD is 3 cm, we first note the coordinates of Point A, which is (-9, 4). Since Point D must have the same second coordinate as Point A, this means that the y-coordinate of Point D must also be 4.

Now, let’s denote Point D's coordinates as (x, 4). We know that the distance AD is given as 3 cm. The distance formula between point A and point D is defined as:

\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

In this case:

\[ \text{Distance} = \sqrt{(x - (-9))^2 + (4 - 4)^2} = \sqrt{(x + 9)^2 + 0} = |x + 9| \]

We want this distance to be equal to 3:

\[ |x + 9| = 3 \]

This absolute value equation gives us two cases:

  1. \( x + 9 = 3 \)
  2. \( x + 9 = -3 \)

For the first case:

\[ x + 9 = 3 \implies x = 3 - 9 \implies x = -6 \]

For the second case:

\[ x + 9 = -3 \implies x = -3 - 9 \implies x = -12 \]

Therefore, Point D could have coordinates (-6, 4) or (-12, 4).

Given the options provided, the only valid choice for Point D that fits the calculated condition is:

(-6, 4)

So the response is:

(−6, 4)