Question

is mnop congruent to wxyz? justify your answer.
yes, because a translation left 4 units and up 9 units followed by a reflection across the x-axis maps mnop onto wxyz.
yes, because a rotation 90° counterclockwise around the origin followed by a translation down 9 units maps mnop onto wxyz.
no, because \\(\overline{pm}\\) and \\(\overline{zw}\\) do not have the same length.
no, because \\(\angle o\\) and \\(\angle y\\) do not have the same measure.

Explanation:

Step1: Analyze Congruence via Transformations

To determine if \( MNOP \) and \( WXYZ \) are congruent, we check if a rigid transformation (translation, reflection, rotation) maps one to the other. Rigid transformations preserve side lengths and angles, so congruent figures can be mapped via such transformations.

Step2: Evaluate Each Option

- **Option 1**: A translation left 4 units and up 9 units, then reflection over \( x \)-axis. Let's check coordinates (e.g., \( M(-2, -2) \), \( W(-6, 7) \)). Translating \( M \) left 4: \( (-6, -2) \), up 9: \( (-6, 7) \), then reflect over \( x \)-axis? Wait, no—wait, reflection over \( x \)-axis would flip \( y \)-sign, but \( W \) has \( y = 7 \). Wait, maybe miscalculation. Wait, actually, let's check side lengths. \( PM \): from \( P(-4, -3) \) to \( M(-2, -2) \): distance \( \sqrt{(-2+4)^2 + (-2+3)^2} = \sqrt{4 + 1} = \sqrt{5} \). \( ZW \): from \( Z(-9, -6) \) to \( W(-7, -8) \): distance \( \sqrt{(-7+9)^2 + (-8+6)^2} = \sqrt{4 + 4} = \sqrt{8} \)? Wait, no, wait coordinates: \( Z \) is at \( (-9, -6) \)? Wait the grid: \( Z \) is at \( x=-9, y=-6 \)? Wait \( W \) is at \( x=-7, y=-8 \). So \( ZW \): \( \Delta x = -7 - (-9) = 2 \), \( \Delta y = -8 - (-6) = -2 \), so distance \( \sqrt{2^2 + (-2)^2} = \sqrt{8} \). \( PM \): \( P(-4, -3) \), \( M(-2, -2) \): \( \Delta x = -2 - (-4) = 2 \), \( \Delta y = -2 - (-3) = 1 \), distance \( \sqrt{2^2 + 1^2} = \sqrt{5} \). Wait, no—wait the third option says \( \overline{PM} \) and \( \overline{ZW} \) have different lengths. Wait, maybe I misread coordinates. Let's re-express coordinates:

- \( M \): \( x=-2, y=-2 \) (since on grid, left 2, up 2 from origin? Wait no, the grid: \( x \)-axis from -10 to 10, \( y \)-axis from -10 to 10. Let's list coordinates:
- \( M \): \( (-2, -2) \) (blue, top of blue quadrilateral)
- \( N \): \( (-1, -5) \) (blue, right)
- \( O \): \( (-4, -7) \) (blue, bottom)
- \( P \): \( (-5, -3) \) (blue, left)
- \( W \): \( (-7, -8) \) (green, bottom)
- \( X \): \( (-6, -4) \) (green, middle)
- \( Y \): \( (-8, -2) \) (green, top)
- \( Z \): \( (-9, -6) \) (green, left)

Now, \( PM \): \( P(-5, -3) \) to \( M(-2, -2) \): \( \Delta x = 3 \), \( \Delta y = 1 \), distance \( \sqrt{3^2 + 1^2} = \sqrt{10} \). \( ZW \): \( Z(-9, -6) \) to \( W(-7, -8) \): \( \Delta x = 2 \), \( \Delta y = -2 \), distance \( \sqrt{2^2 + (-2)^2} = \sqrt{8} \). Wait, no, that's not matching. Wait maybe the third option is wrong? Wait no, wait the third option says "No, because \( \overline{PM} \) and \( \overline{ZW} \) do not have the same length." Wait, let's recalculate \( PM \) and \( ZW \):

\( P(-4, -3) \)? Wait maybe I misread the \( x \)-coordinates. Let's check the grid: \( x \)-axis labels: -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 1, 2,... So \( M \) is at \( x=-2 \) (between -3 and -1), \( y=-2 \) (between -3 and -1). \( P \) is at \( x=-4 \) (between -5 and -3), \( y=-3 \) (between -4 and -2). So \( PM \): \( x \)-difference: \( -2 - (-4) = 2 \), \( y \)-difference: \( -2 - (-3) = 1 \), so length \( \sqrt{2^2 + 1^2} = \sqrt{5} \).

\( Z \) is at \( x=-9 \) (leftmost green), \( y=-6 \) (between -7 and -5). \( W \) is at \( x=-7 \) (right of \( Z \)), \( y=-8 \) (between -9 and -7). So \( ZW \): \( x \)-difference: \( -7 - (-9) = 2 \), \( y \)-difference: \( -8 - (-6) = -2 \), length \( \sqrt{2^2 + (-2)^2} = \sqrt{8} \). Wait, these are different. But wait, maybe my coordinate reading is wrong. Wait the blue figure \( MNOP \): \( M \) at \( (-2, -2) \), \( N(-1, -5) \), \( O(-4, -7) \), \( P(-5, -3) \). Green figure \( WXYZ \): \( W(-7, -8) \), \( X(-6, -4) \), \( Y(-8, -2) \), \( Z(-9, -6) \).

Now, \( PM \): from \( P(-5, -3) \) to \( M(-2, -2) \): \( \Delta x = 3 \), \( \Delta y = 1 \), length \( \sqrt{9 + 1} = \sqrt{10} \). \( ZW \): from \( Z(-9, -6) \) to \( W(-7, -8) \): \( \Delta x = 2 \), \( \Delta y = -2 \), length \( \sqrt{4 + 4} = \sqrt{8} \). Wait, these are different. But wait, the first option's transformation: translating \( M(-2, -2) \) left 4: \( -6, -2 \), up 9: \( -6, 7 \), then reflect over \( x \)-axis? No, \( W \) is at \( (-7, -8) \)? Wait, I'm confused. Wait maybe the correct answer is the third option? Wait no, the third option says "No, because \( \overline{PM} \) and \( \overline{ZW} \) do not have the same length." Let's check another side. \( MN \): \( M(-2, -2) \) to \( N(-1, -5) \): \( \Delta x = 1 \), \( \Delta y = -3 \), length \( \sqrt{1 + 9} = \sqrt{10} \). \( WX \): \( W(-7, -8) \) to \( X(-6, -4) \): \( \Delta x = 1 \), \( \Delta y = 4 \), length \( \sqrt{1 + 16} = \sqrt{17} \). Wait, no, that can't be. Wait maybe I messed up the figures. Wait the blue figure is \( MNOP \), green is \( WXYZ \). Let's check the first option's transformation again. Suppose we take \( M(-2, -2) \), translate left 4: \( x=-6 \), up 9: \( y=7 \), so \( (-6, 7) \). Then reflect over \( x \)-axis: \( y \) becomes \( -7 \), but \( W \) is at \( (-7, -8) \). No, that's not matching. Wait the second option: rotation 90° counterclockwise (which would map \( (x,y) \) to \( (-y, x) \)) then translation down 9. Let's take \( M(-2, -2) \): rotate 90° CCW: \( (2, -2) \), then translate down 9: \( (2, -11) \), which is not \( W \). So that's wrong.

Wait the third option: "No, because \( \overline{PM} \) and \( \overline{ZW} \) do not have the same length." Let's calculate \( PM \) and \( ZW \) correctly. \( P(-4, -3) \), \( M(-2, -2) \): \( \Delta x = 2 \), \( \Delta y = 1 \), length \( \sqrt{4 + 1} = \sqrt{5} \). \( Z(-9, -6) \), \( W(-7, -8) \): \( \Delta x = 2 \), \( \Delta y = -2 \), length \( \sqrt{4 + 4} = \sqrt{8} \). These are different, so \( PM \) and \( ZW \) (corresponding sides) have different lengths, so the figures can't be congruent. The fourth option: \( \angle O \) and \( \angle Y \)—but if sides are different, angles would be too, but the third option directly addresses side lengths.

Answer:

No, because \( \overline{PM} \) and \( \overline{ZW} \) do not have the same length. (The third option: "No, because \( \overline{PM} \) and \( \overline{ZW} \) do not have the same length.")