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Which of the following statements about EF and EG is true?

An image displaying a graph plotted by two lines EF with coordinates E of -4 comma -4 and F of 2 comma -3 and EG with two coordinates E of -4 comma -4 and G of -9 comma -2. Which of the following statements is true about the lines EF and EG?
A.
EF and EG are the same length.
B.
EG is longer than EF.
C.
EF is longer than EG.
D.
EF is 7 units long.

1 answer

To determine the relationship between the lengths of lines EF and EG, we need to calculate their lengths using the distance formula, which is:

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

  1. Calculate the length of line EF:

    • Coordinates of E: \((-4, -4)\)
    • Coordinates of F: \((2, -3)\)

    Using the distance formula:

    \[ EF = \sqrt{(2 - (-4))^2 + (-3 - (-4))^2} \] \[ EF = \sqrt{(2 + 4)^2 + (-3 + 4)^2} \] \[ EF = \sqrt{(6)^2 + (1)^2} \] \[ EF = \sqrt{36 + 1} \] \[ EF = \sqrt{37} \]

  2. Calculate the length of line EG:

    • Coordinates of E: \((-4, -4)\)
    • Coordinates of G: \((-9, -2)\)

    Using the distance formula:

    \[ EG = \sqrt{(-9 - (-4))^2 + (-2 - (-4))^2} \] \[ EG = \sqrt{(-9 + 4)^2 + (-2 + 4)^2} \] \[ EG = \sqrt{(-5)^2 + (2)^2} \] \[ EG = \sqrt{25 + 4} \] \[ EG = \sqrt{29} \]

  3. Compare lengths:

    • \(EF = \sqrt{37}\)
    • \(EG = \sqrt{29}\)

To compare \( \sqrt{37} \) and \( \sqrt{29} \):

Since \(37 > 29\), it follows that \( \sqrt{37} > \sqrt{29} \).

Thus, \(EF\) is longer than \(EG\).

Now let's look at the provided answer choices:

  • A. EF and EG are the same length. (False)
  • B. EG is longer than EF. (False)
  • C. EF is longer than EG. (True)
  • D. EF is 7 units long. (False)

Therefore, the true statement is:

C. EF is longer than EG.