To assess whether the parallelograms ADCB and EHGF are congruent, we need to determine if one can be transformed into the other through rigid transformations such as reflection, rotation, or translation. Dilations, on the other hand, impact the size and therefore would not preserve congruence.
Let's analyze the coordinates of both parallelograms:
Parallelogram ADCB:
- A: (-6, 1)
- B: (-3, 1)
- C: (-2, 3)
- D: (-5, 3)
Parallelogram EHGF:
- E: (2, 3)
- H: (3, 5)
- G: (6, 5)
- F: (5, 3)
Step 1: Calculate the lengths of the sides and check for congruence
Parallelogram ADCB:
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AB = distance between A and B: \[ AB = \sqrt{((-3) - (-6))^2 + (1 - 1)^2} = \sqrt{(3)^2} = 3 \]
-
AD = distance between A and D: \[ AD = \sqrt{((-5) - (-6))^2 + (3 - 1)^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \]
-
BC = distance between B and C: \[ BC = \sqrt{((-2) - (-3))^2 + (3 - 1)^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{5} \]
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CD = distance between C and D: \[ CD = \sqrt{((-5) - (-2))^2 + (3 - 3)^2} = \sqrt{(-3)^2} = 3 \]
The lengths of the sides of ADCB are:
- AB = 3
- AD = √5
- BC = √5
- CD = 3
Parallelogram EHGF:
-
EH = distance between E and H: \[ EH = \sqrt{((3) - (2))^2 + (5 - 3)^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{5} \]
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EG = distance between E and G: \[ EG = \sqrt{((6) - (2))^2 + (5 - 3)^2} = \sqrt{(4)^2 + (2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]
-
HF = distance between H and F: \[ HF = \sqrt{((5) - (3))^2 + (3 - 5)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
-
GF = distance between G and F: \[ GF = \sqrt{((5) - (6))^2 + (3 - 5)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]
The lengths of the sides are:
- EH = √5
- EG = 2√5
- HF = 2√2
- GF = √5
Step 2: Compare side lengths
From the side lengths:
- Parallelogram ADCB has two sides of length 3 and two sides of length √5.
- Parallelogram EHGF has side lengths that do not match those of ADCB, hence they are not congruent.
Conclusion
Since the lengths of sides in both parallelograms do not match, they cannot be congruent. The option that reflects this conclusion is:
C. No, ADCB is not congruent to EHGF because parallelogram ADCB was dilated.
However, it should also be noted that the statement about dilation is incorrect, but the main point is that they are not congruent due to the difference in side lengths.