To determine if the parallelograms ADCB and EHGF are congruent, we need to compare their properties, specifically their side lengths and angles. Two shapes are congruent if one can be transformed into the other by rigid motions such as translation, rotation, or reflection, without changing their size or shape.
First, let's compute the lengths of the sides of both parallelograms.
Parallelogram ADCB:
- A(-6,1) to B(-3,1):
Length = √((-3 - (-6))^2 + (1 - 1)^2) = √(3^2 + 0^2) = 3 - B(-3,1) to C(-2,3):
Length = √((-2 - (-3))^2 + (3 - 1)^2) = √(1^2 + 2^2) = √(1 + 4) = √5 - C(-2,3) to D(-5,3):
Length = √((-5 - (-2))^2 + (3 - 3)^2) = √((-3)^2 + 0^2) = 3 - D(-5,3) to A(-6,1):
Length = √((-6 - (-5))^2 + (1 - 3)^2) = √((-1)^2 + (-2)^2) = √(1 + 4) = √5
Thus, the sides of parallelogram ADCB are 3, √5, 3, and √5.
Parallelogram EHGF:
- E(2,3) to H(3,5):
Length = √((3 - 2)^2 + (5 - 3)^2) = √(1^2 + 2^2) = √(1 + 4) = √5 - H(3,5) to G(6,5):
Length = √((6 - 3)^2 + (5 - 5)^2) = √(3^2 + 0^2) = 3 - G(6,5) to F(5,3):
Length = √((5 - 6)^2 + (3 - 5)^2) = √((-1)^2 + (-2)^2) = √(1 + 4) = √5 - F(5,3) to E(2,3):
Length = √((2 - 5)^2 + (3 - 3)^2) = √((-3)^2 + 0^2) = 3
Thus, the sides of parallelogram EHGF are √5, 3, √5, and 3.
Conclusion:
Since both parallelograms ADCB and EHGF have the same side lengths of 3 and √5, they are congruent.
Now looking at the options provided:
- a. Yes, ADCB≅EHGF because parallelogram ADCB was reflected
- b. No, ADCB is not congruent to EHGF because parallelogram ADCB was dilated
- c. Yes, ADCB≅EHGF because parallelogram ADCB was rotated
- d. Yes, ADCB≅EHGF since parallelogram ADCB was translated
The most fitting option, based on congruency through movements like reflection, rotation, or translation, is:
a. Yes, ADCB≅EHGF because parallelogram ADCB was reflected.
However, it's equally valid to say they might be congruent through any of the mentioned rigid motions, but without knowing the exact transformation in the context, option (a) is a reasonable choice.
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