Determine whether the polygons with the given vertices are similar.

Quadrilateral ABCD with vertices A(–3, 3), B(0, 3), C(0, 1), D(–3, 1)
and quadrilateral EFGH with vertices E(2, –8), F(8, –8), G(8, –12), H(2, –12)

a little inspection (or drawing) will reveal that

ABCD is a rectangle with width=3 and height=2

EFGH is a rectangle with width=6 and height=4

Since the width and height have both been scaled by 2, the two rectangles are similar.

24

To determine whether two polygons are similar, we need to compare their corresponding sides and angles.

Let's start by comparing the corresponding sides of the two quadrilaterals:

Quadrilateral ABCD:
Side AB has length of 0 - (-3) = 3 units.
Side BC has length of 1 - 3 = -2 units.
Side CD has length of -3 - 0 = -3 units.
Side DA has length of 1 - 3 = -2 units.

Quadrilateral EFGH:
Side EF has length of 8 - 2 = 6 units.
Side FG has length of -12 - (-8) = -4 units.
Side GH has length of 2 - 8 = -6 units.
Side HE has length of -12 - (-8) = -4 units.

Now, let's compare the corresponding angles of the two quadrilaterals:

Quadrilateral ABCD:
Angle A measures (0, 3), B(0, 3), C(0, 1): This angle is a 90-degree angle.
Angle B measures (0, 3), C(0, 1), D(-3, 1): This angle is a 90-degree angle.
Angle C measures C(0, 1), D(-3, 1), A(-3, 3): This angle is a 90-degree angle.
Angle D measures D(-3, 1), A(-3, 3), B(0, 3): This angle is a 90-degree angle.

Quadrilateral EFGH:
Angle E measures (8, -8), F(8, -8), G(8, -12): This angle is a 90-degree angle.
Angle F measures (8, -8), G(8, -12), H(2, -12): This angle is a 90-degree angle.
Angle G measures G(8, -12), H(2, -12), E(2, -8): This angle is a 90-degree angle.
Angle H measures H(2, -12), E(2, -8), F(8, -8): This angle is a 90-degree angle.

Since the corresponding sides are proportional and the corresponding angles are equal (all 90-degree angles), we can conclude that the quadrilateral ABCD and quadrilateral EFGH are similar.