Assess parallelograms A(-6,1)B(-3,1)C(-2,3)D(-5,3) and EHGF to decide if they are congruent.

Question

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

Answer 1

To assess whether parallelogram ABCD and parallelogram EHGF are congruent, we need to analyze the properties of the given points and the transformations that can prove congruency.

Firstly, let's verify the properties of parallelogram ABCD:

• The coordinates of the vertices are:
  • A(-6, 1)
  • B(-3, 1)
  • C(-2, 3)
  • D(-5, 3)

We can calculate the lengths of the sides AB, BC, CD, and DA to ascertain if it's a parallelogram and check for congruency.

1. Length of AB: \[ AB = \sqrt{(-3 - (-6))^2 + (1 - 1)^2} = \sqrt{(3)^2 + 0} = 3 \]

2. Length of BC: \[ BC = \sqrt{(-2 - (-3))^2 + (3 - 1)^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{5} \]

3. Length of CD: \[ CD = \sqrt{(-5 - (-2))^2 + (3 - 3)^2} = \sqrt{(-3)^2 + 0} = 3 \]

4. Length of DA: \[ DA = \sqrt{(-6 - (-5))^2 + (1 - 3)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{5} \]

Now we see that:

• AB = CD = 3
• BC = DA = √5

Since both pairs of opposite sides are equal, ABCD is indeed a parallelogram.

Next, we assess whether parallelograms ABCD and EHGF are congruent based on some transformation.

The options provided suggest the possibility of reflection, dilation, rotation, or translation. Two figures are congruent if one can be obtained from the other through these transformations without changing their size or shape.

To determine if the statements a, b, c, or d are correct, we can say:

• If parallelogram ABCD was reflected, rotated, or translated without changing its dimensions, it would still remain congruent.
• Dilation, however, does change the dimensions and angles of the figures.

Since the problem doesn't provide specific coordinates or transformations for parallelogram EHGF, we cannot determine its dimensions and angles directly.

Given these considerations, without having any conflicting dilation transformation, the most logical choice would be:

d. Yes, ADCB≅EHGF since parallelogram ADCB was translated.

This assumes that EHGF behaves correspondingly to the properties of ABCD through translation or other congruence-preserving transformations. Without knowing its specific measurements or transformations in the problem, the other options referencing dilation would not be valid.