the graph shows quadrilaterals cdef and rstu.
is cdef congruent to rstu? justify your answer.
yes, because a reflection across the y - axis maps cdef onto rstu.
yes, because a rotation 90° clockwise around the origin maps cdef onto rstu.
no, because $\overline{fc}$ and $\overline{ur}$ do not have the same length.
no, because $\overline{de}$ and $\overline{st}$ do not have the same length.

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# Brief Explanations:
To determine if CDEF and RSTU are congruent, we can check the lengths of corresponding sides. Let's find the coordinates of the vertices:
- For CDEF: Let's assume coordinates (from the graph, approximate): C(4, -9), D(8, -5), E(6, -2), F(6, -5)
- For RSTU: S(-8, -5), T(-6, -2), U(-6, -5), R(-4, -9)

Now, calculate the length of $ \overline{DE} $ (in CDEF) and $ \overline{ST} $ (in RSTU) using the distance formula $ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $ or by counting grid units (since horizontal/vertical lines).

For $ \overline{DE} $: From D(8, -5) to E(6, -2). The horizontal distance is $ 8 - 6 = 2 $, vertical distance is $ -2 - (-5) = 3 $. So length is $ \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} $.

For $ \overline{ST} $: From S(-8, -5) to T(-6, -2). Horizontal distance $ -6 - (-8) = 2 $, vertical distance $ -2 - (-5) = 3 $. Length is $ \sqrt{2^2 + 3^2} = \sqrt{13} $? Wait, no, wait. Wait, maybe I made a mistake. Wait, let's check another side. Wait, actually, let's check $ \overline{DE} $ and $ \overline{ST} $ again. Wait, no, the fourth option says "No, because $ \overline{DE} $ and $ \overline{ST} $ do not have the same length." Wait, maybe my coordinate assumption is wrong. Let's re - examine the graph.

Looking at the vertical lines: For $ \overline{EF} $ in CDEF: E(6, -2) to F(6, -5). That's a vertical line, length is $ |-2 - (-5)| = 3 $. For $ \overline{TU} $ in RSTU: T(-6, -2) to U(-6, -5). Vertical line, length is $ |-2 - (-5)| = 3 $. Now, $ \overline{DE} $: D(8, -5) to E(6, -2). The change in x: $ 8 - 6 = 2 $, change in y: $ -5 - (-2) = -3 $, so length $ \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.605 $.

$ \overline{ST} $: S(-8, -5) to T(-6, -2). Change in x: $ -6 - (-8) = 2 $, change in y: $ -2 - (-5) = 3 $, length $ \sqrt{(2)^2 + (3)^2} = \sqrt{13} \approx 3.605 $. Wait, that's the same. Wait, maybe another side. Wait, $ \overline{FC} $ and $ \overline{UR} $. $ \overline{FC} $: F(6, -5) to C(4, -9). Change in x: $ 4 - 6 = -2 $, change in y: $ -9 - (-5) = -4 $, length $ \sqrt{(-2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \approx 4.472 $. $ \overline{UR} $: U(-6, -5) to R(-4, -9). Change in x: $ -4 - (-6) = 2 $, change in y: $ -9 - (-5) = -4 $, length $ \sqrt{(2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \approx 4.472 $. Wait, maybe the mistake is in the first two options. Let's check the reflection. A reflection across the y - axis would map (x, y) to (-x, y). Let's take point C(4, -9): reflection over y - axis is (-4, -9), which is R. F(6, -5): reflection is (-6, -5), which is U. E(6, -2): reflection is (-6, -2), which is T. D(8, -5): reflection is (-8, -5), which is S. Wait, that's exactly the mapping! So a reflection across the y - axis maps C(4, -9) to R(-4, -9), F(6, -5) to U(-6, -5), E(6, -2) to T(-6, -2), D(8, -5) to S(-8, -5). So the first option says "Yes, because a reflection across the y - axis maps CDEF onto RSTU." But wait, let's check the sides again. Wait, maybe I messed up the coordinates. Let's list the coordinates properly:

- CDEF:
  - C: Let's say from the graph, C is at (4, -9) (since it's 4 units right on x - axis from origin, 9 units down on y - axis)
  - D: (8, -5) (8 right, 5 down)
  - E: (6, -2) (6 right, 2 down)
  - F: (6, -5) (6 right, 5 down)

- RSTU:
  - R: (-4, -9) (4 left, 9 down)
  - S: (-8, -5) (8 left, 5 down)
  - T: (-6, -2) (6 left, 2 down)
  - U: (-6, -5) (6 left, 5 down)

Now, reflecting CDEF over the y - axis (rule: (x, y) → (-x, y)):

- C(4, -9) → (-4, -9) = R
- D(8, -5) → (-8, -5) = S
- E(6, -2) → (-6, -2) = T
- F(6, -5) → (-6, -5) = U

So the reflection across the y - axis maps C(4, -9) to R(-4, -9), D(8, -5) to S(-8, -5), E(6, -2) to T(-6, -2), F(6, -5) to U(-6, -5). So this reflection maps quadrilateral CDEF onto quadrilateral RSTU. Since a reflection is a rigid transformation (preserves distance and angle), the two quadrilaterals are congruent.

Now, let's check the other options:

- Option 2: Rotation 90° clockwise around origin. The rule for 90° clockwise rotation is (x, y) → (y, -x). Let's test C(4, -9): ( - 9, - 4), which is not R(-4, -9). So this is incorrect.
- Option 3: $ \overline{FC} $ and $ \overline{UR} $: $ \overline{FC} $ length: distance between F(6, -5) and C(4, -9): $ \sqrt{(4 - 6)^2+( - 9+5)^2}=\sqrt{(-2)^2+( - 4)^2}=\sqrt{4 + 16}=\sqrt{20} $. $ \overline{UR} $: distance between U(-6, -5) and R(-4, -9): $ \sqrt{(-4 + 6)^2+( - 9 + 5)^2}=\sqrt{(2)^2+( - 4)^2}=\sqrt{4 + 16}=\sqrt{20} $. So they have the same length. So this option is wrong.
- Option 4: $ \overline{DE} $ and $ \overline{ST} $: $ \overline{DE} $: distance between D(8, -5) and E(6, -2): $ \sqrt{(6 - 8)^2+( - 2 + 5)^2}=\sqrt{(-2)^2+(3)^2}=\sqrt{4 + 9}=\sqrt{13} $. $ \overline{ST} $: distance between S(-8, -5) and T(-6, -2): $ \sqrt{(-6 + 8)^2+( - 2 + 5)^2}=\sqrt{(2)^2+(3)^2}=\sqrt{4 + 9}=\sqrt{13} $. So they have the same length. So this option is wrong.

So the correct answer is the first option.
# Answer:
Yes, because a reflection across the y - axis maps CDEF onto RSTU.

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