Question

Coordinate plane with quadrilaterals ABCD and EFGH with A at 0 comma 0, B at 2 comma 0, C at 3 comma negative 2, D at negative 1 comma negative 2, E at 2 comma 4, F at 6 comma 4, G at 7 comma 2, and H at 1 comma 2

Are quadrilaterals ABCD and EFGH similar?

Yes, quadrilaterals ABCD and EFGH are similar because a translation of (x + 2, y + 4) and a dilation by the scale factor of 2 from point D′ map quadrilateral ABCD onto EFGH.
Yes, quadrilaterals ABCD and EFGH are similar because a translation of (x + 3, y + 4) and a dilation by the scale factor of 2 from point A′ map quadrilateral ABCD onto EFGH.
No, quadrilaterals ABCD and EFGH are not similar because their corresponding angles are not congruent.
No, quadrilaterals ABCD and EFGH are not similar because their corresponding segments are not proportional.

Answer 1

The answer is: No, quadrilaterals ABCD and EFGH are not similar because their corresponding segments are not proportional.

To determine if quadrilaterals ABCD and EFGH are similar, we need to check if their corresponding angles are congruent and their corresponding sides are proportional.

Verifying Corresponding Angles:

∠A = ∠E (both are right angles)

∠B = ∠F (both are acute angles)

∠C = ∠G (both are obtuse angles)

∠D = ∠H (both are acute angles)

Since all corresponding angles are congruent, we can conclude that quadrilaterals ABCD and EFGH have similar angles.

Verifying Corresponding Sides:

We can calculate the ratios of corresponding side lengths:

AB/EF = 2/2 = 1

BC/FG = 3/6 = 1/2

CD/GH = (-2)/2 = -1

AD/EH = (-1)/1 = -1

As we can see, the ratios of corresponding side lengths are not all equal, specifically BC/FG and CD/GH are not equal to 1. Therefore, quadrilaterals ABCD and EFGH are not proportional in their side lengths.

Conclusion:

While quadrilaterals ABCD and EFGH have similar angles, they are not similar in their side lengths. Therefore, the answer is: No, quadrilaterals ABCD and EFGH are not similar because their corresponding segments are not proportional.