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

Point D
and Point F
have the same second coordinate, and ECFD
is a quadrilateral. What coordinates would Point F
be plotted at to make line segment FD
have a length of 2 cm?

(1 point)
Responses

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

(−1, 1)
left parenthesis negative 1 comma 1 right parenthesis

(1, −1)
left parenthesis 1 comma negative 1 right parenthesis

(−1, 0)

1 answer

To find the coordinates of Point F such that line segment FD has a length of 2 cm and shares the same second coordinate as Point D, we have the following:

  • The coordinates of Point D are \((-3, 1)\).
  • Since Point F must have the same second coordinate as D, its y-coordinate will also be 1, so we can denote Point F as \((x_F, 1)\).

To find the length of line segment FD, we use the distance formula, which in this case simplifies to calculating the absolute difference in the x-coordinates, since the y-coordinates are the same:

\[ \text{Length of } FD = |x_F - (-3)| = |x_F + 3| \]

We want this length to equal 2, so we set up the equation:

\[ |x_F + 3| = 2 \]

This absolute value equation gives us two cases to solve:

  1. Case 1:
    \(x_F + 3 = 2\)
    \(x_F = 2 - 3\)
    \(x_F = -1\)

  2. Case 2:
    \(x_F + 3 = -2\)
    \(x_F = -2 - 3\)
    \(x_F = -5\)

From these cases, we find two possible coordinates for Point F:

  • From Case 1: \(F = (-1, 1)\)
  • From Case 2: \(F = (-5, 1)\) (This point is not among the response options.)

Thus, the coordinates of Point F that make the length of FD equal to 2 cm is \((-1, 1)\).

The correct response is: \[ (-1, 1) \quad \text{(which matches the choice "(-1, 1)") } \]