Question

use the drop-down menus to explain if the two figures below are congruent, similar, or neither. if the figures are similar, state the scale factor.

Explanation:

Step1: Identify Coordinates

Find coordinates of vertices for both trapezoids.
Top trapezoid (J, I, H, G):
- \( J(3,6) \), \( I(4,2) \), \( H(2,2) \), \( G(2,6) \)
Bottom trapezoid (D, C, B, E):
- \( D(8,4) \), \( C(4,4) \), \( B(4, -4) \), \( E(6, -4) \) Wait, correction: Recheck grid. Assume y-axis and x-axis labels (maybe flipped? Let's re-express with correct grid. Let's take horizontal as x, vertical as y.

Correct coordinates (assuming grid lines: each square is 1 unit):
Top trapezoid (small):
- \( J(3,6) \)? No, looking at the graph, top trapezoid has vertices:
\( J(3, 6) \)? Wait, the top trapezoid: G(2,6), H(2,2), I(4,2), J(3,6)? Wait, no, let's count grid. Let's see:
G is at (2,6), H at (2,2), I at (4,2), J at (3,6)? Wait, no, J should be at (3,6)? Wait, maybe better to find side lengths.

Top trapezoid (small):
- Horizontal side (GH or JI? Wait, GH: from (2,6) to (2,2): length \( 6 - 2 = 4 \)? No, vertical distance: \( |6 - 2| = 4 \)? Wait, no, horizontal and vertical. Wait, GH is vertical: x=2, y from 2 to 6: length 4? No, y from 2 to 6: 4 units? Wait, no, 6 - 2 = 4? Wait, no, 6 - 2 = 4? Wait, 6 - 2 is 4, so length 4. Then HI: from (2,2) to (4,2): horizontal, length \( 4 - 2 = 2 \). Then IJ: from (4,2) to (3,6): slope, but maybe better to find the two parallel sides (bases) of the trapezoid.

Wait, trapezoid has two parallel sides (bases). Let's find the lengths of the two bases for both trapezoids.

Small trapezoid (top):
- Lower base (HI): from (2,2) to (4,2): length \( 4 - 2 = 2 \).
- Upper base (JG): from (2,6) to (3,6)? No, wait, J is at (3,6), G at (2,6): length \( 3 - 2 = 1 \)? No, that can't be. Wait, maybe I misread the graph. Let's look again: the top trapezoid: G(2,6), H(2,2), I(4,2), J(3,6). So:
- GH: vertical, from (2,6) to (2,2): length \( 6 - 2 = 4 \) (vertical side).
- HI: horizontal, from (2,2) to (4,2): length \( 4 - 2 = 2 \) (lower base).
- IJ: from (4,2) to (3,6): slope, but maybe the other base: JG: from (3,6) to (2,6): length \( 3 - 2 = 1 \)? No, that's not parallel. Wait, no, trapezoid has two parallel sides (bases) which are horizontal. So HI is horizontal (y=2), length 2. JG: is it horizontal? J(3,6), G(2,6): yes, y=6, length \( 3 - 2 = 1 \)? No, that's 1. Wait, that can't be. Maybe the top trapezoid has bases of length 1 and 2, and the bottom trapezoid has bases of length 2 and 4? Wait, no, let's check the bottom trapezoid.

Bottom trapezoid (large):
- Lower base (BE): from (4, -4) to (6, -4): length \( 6 - 4 = 2 \)? No, wait, no, let's re-express with correct axis. Let's assume the vertical axis is y, horizontal is x. The bottom trapezoid: C(4,4), B(4, -4), E(6, -4), D(8,4). Wait, no, looking at the graph, the bottom trapezoid has vertices: C(4,4), B(4, -4), E(6, -4), D(8,4). So:
- CB: vertical, from (4,4) to (4, -4): length \( 4 - (-4) = 8 \).
- BE: horizontal, from (4, -4) to (6, -4): length \( 6 - 4 = 2 \).
- ED: from (6, -4) to (8,4): slope, but upper base: CD: from (4,4) to (8,4): length \( 8 - 4 = 4 \).

Wait, now small trapezoid:
- Vertical side (GH): from (2,6) to (2,2): length \( 6 - 2 = 4 \).
- Lower base (HI): from (2,2) to (4,2): length \( 4 - 2 = 2 \).
- Upper base (JG): from (2,6) to (3,6)? No, J is at (3,6), G at (2,6): length 1? No, that's wrong. Wait, maybe the top trapezoid has upper base length 1 and lower base length 2, and the bottom trapezoid has upper base length 4 and lower base length 8? Wait, no, let's check the ratio.

Wait, maybe the small trapezoid has bases of length 1 and 2, and the large one has bases of length 2 and 4? No, that would be scale factor 2. Wait, let's find the length of the vertical sides (the legs) and the horizontal bases.

Small trapezoid (top):
- Vertical leg (GH): from (2,6) to (2,2): length \( 6 - 2 = 4 \)? No, 6 - 2 is 4? Wait, 6 - 2 = 4, so length 4.
- Horizontal base (HI): from (2,2) to (4,2): length \( 4 - 2 = 2 \).
- Upper base (JG): from (3,6) to (2,6): length \( 3 - 2 = 1 \)? No, that's 1. Wait, no, J should be at (3,6), G at (2,6): length 1. Then the other base (HI) is 2. So the two bases are 1 and 2, leg length 4? No, that doesn't make sense. Wait, maybe I flipped x and y. Let's assume the horizontal axis is y, vertical is x. So the coordinates are (y, x). Then:

Top trapezoid:
- G(6,2), H(2,2), I(2,4), J(6,3)? No, this is confusing. Let's use the side lengths.

Alternative approach: The small trapezoid has a horizontal side (the bottom base) of length 2 (from x=2 to x=4, y=2) and the top base (from x=2 to x=3, y=6) of length 1? No, that's not parallel. Wait, no, trapezoid must have two parallel sides. So the two parallel sides are the vertical sides? No, GH and IJ: GH is vertical (x=2, y from 2 to 6), IJ is from (4,2) to (3,6): not vertical. HI and JG: HI is horizontal (y=2, x from 2 to 4), JG is horizontal (y=6, x from 2 to 3)? No, x from 2 to 3 is length 1. So HI length 2, JG length 1. Then the legs are GH (length 4) and IJ (length \( \sqrt{(4-3)^2 + (2-6)^2} = \sqrt{1 + 16} = \sqrt{17} \))? No, that can't be similar. Wait, maybe the bottom trapezoid: let's find its side lengths.

Bottom trapezoid (large):
- Horizontal side (BE): from x=4 to x=6, y=-4: length 2.
- Horizontal side (CD): from x=4 to x=8, y=4: length 4.
- Vertical side (CB): from x=4, y=4 to x=4, y=-4: length 8.
- Vertical side (ED): from x=6, y=-4 to x=8, y=4: length \( \sqrt{(8-6)^2 + (4 - (-4))^2} = \sqrt{4 + 64} = \sqrt{68} = 2\sqrt{17} \).

Wait, small trapezoid:
- Horizontal sides: 1 and 2 (JG=1, HI=2).
- Vertical sides: GH=4 (y from 2 to 6, x=2), IJ: \( \sqrt{(4-3)^2 + (2-6)^2} = \sqrt{17} \).

Large trapezoid:
- Horizontal sides: 2 and 4 (BE=2, CD=4).
- Vertical sides: CB=8 (y from -4 to 4, x=4), ED: \( \sqrt{(8-6)^2 + (4 - (-4))^2} = \sqrt{4 + 64} = \sqrt{68} = 2\sqrt{17} \).

Now, check ratios:
- Horizontal sides: \( \frac{2}{1} = 2 \), \( \frac{4}{2} = 2 \).
- Vertical sides: \( \frac{8}{4} = 2 \), \( \frac{2\sqrt{17}}{\sqrt{17}} = 2 \).

Ah, so the scale factor is 2 (large to small: 2, small to large: 1/2? Wait, no: small trapezoid has horizontal sides 1 and 2, large has 2 and 4. So ratio of large to small is \( \frac{2}{1} = 2 \) and \( \frac{4}{2} = 2 \). So the scale factor from small to large is 2.

Step2: Confirm Similarity

Since all corresponding sides are in ratio 2 (large trapezoid sides are twice the small trapezoid sides), the figures are similar, and the scale factor (small to large) is 2.

Answer:

The two figures are similar, and the scale factor is \( \boldsymbol{2} \) (from the small trapezoid to the large trapezoid) or \( \frac{1}{2} \) (large to small, but typically scale factor is from original to image; here small is original, large is image, so scale factor 2).