Question

the graph shows quadrilaterals defg and nopq.
is defg congruent to nopq? justify your answer.
yes, because a reflection across the y - axis maps defg onto nopq.
yes, because a translation right 12 units maps defg onto nopq.
no, because \\(\overline{ef}\\) and \\(\overline{op}\\) do not have the same length.
no, because \\(\overline{gd}\\) and \\(\overline{qn}\\) do not have the same length.

Explanation:

Step1: Identify coordinates of vertices

First, find the coordinates of each vertex of both quadrilaterals. For DEFG:

• \( D(-6, 2) \)
• \( E(-8, 5) \)
• \( F(-5, 8) \)
• \( G(-3, 5) \)

For NOPQ:

• \( N(6, 2) \)
• \( O(4, 5) \)
• \( P(7, 8) \)
• \( Q(9, 5) \)

Step2: Analyze translation

A translation right 12 units means adding 12 to the x - coordinate of each vertex of DEFG. Let's check:

• For \( D(-6, 2) \): \( -6 + 12 = 6 \), so \( (6, 2) \) which is \( N \).
• For \( E(-8, 5) \): \( -8 + 12 = 4 \), so \( (4, 5) \) which is \( O \).
• For \( F(-5, 8) \): \( -5 + 12 = 7 \), so \( (7, 8) \) which is \( P \).
• For \( G(-3, 5) \): \( -3 + 12 = 9 \), so \( (9, 5) \) which is \( Q \).

Since a translation (a rigid transformation) maps DEFG onto NOPQ, and rigid transformations preserve congruence, the quadrilaterals are congruent.

Now, let's analyze the other options:

• Option 1: A reflection across the y - axis would change the sign of the x - coordinate. For \( D(-6,2) \), reflection over y - axis is \( (6,2) \) (which is N), but for \( E(-8,5) \), reflection over y - axis is \( (8,5) \), not \( O(4,5) \). So this is incorrect.
• Option 3: Calculate the length of \( \overline{EF} \) and \( \overline{OP} \). Using the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). For \( EF \): \( \sqrt{(-5+8)^2+(8 - 5)^2}=\sqrt{9 + 9}=\sqrt{18} \). For \( OP \): \( \sqrt{(7 - 4)^2+(8 - 5)^2}=\sqrt{9 + 9}=\sqrt{18} \). So they have the same length, so this option is incorrect.
• Option 4: Calculate the length of \( \overline{GD} \) and \( \overline{QN} \). For \( GD \): \( \sqrt{(-3 + 6)^2+(5 - 2)^2}=\sqrt{9+9}=\sqrt{18} \). For \( QN \): \( \sqrt{(9 - 6)^2+(5 - 2)^2}=\sqrt{9 + 9}=\sqrt{18} \). So they have the same length, so this option is incorrect.

Answer:

Yes, because a translation right 12 units maps DEFG onto NOPQ.