Question

A gridded coordinate plane displays a light-blue shaded triangle, ABC, superimposed on top of a slightly larger light-blue shaded triangle, CDE. The vertices of the smaller triangle are located at approximately (3,2), (3,6), and (9,6). The vertices of the larger triangle are located at approximately (9,8), (9,6), and (12,9). The vertices of both triangles are marked with black dots. A line, also in black, passes through the points (1, 0) and (12, 9), forming the hypotenuse of both the smaller and larger triangles. The coordinates and vertices of the triangles are clearly labeled.


Triangles ABC and CDE are called Response area triangles.

The slope of the line through AE can be written as Response area

This is a Response area relationship because Response area

The equation of the line AE is written as Response area

options for response areas:
Similar
Congruent
Disproportional
2/3
3/2
2
3
Nonproportional
Proportional
Non Linear
There is a constant rate of change and it goes through (0, 0)
it shows a linear relationship
the graph is increasing
the triangles are both the same
y = 3/2x
y = 2/3 x + 1
y = 2/3x
y = 2x + 3
2y = 3x

Answer 1

Given the information about triangles ABC and CDE, we'll first determine the slope of line AE, which could be helpful to categorize the relationship and to determine the equation of the line.

### Step 1: Find the slope of line AE

Let's use points A (3, 6) and E (12, 9) to find the slope.

The slope \( m \) between two points (x1, y1) and (x2, y2) is given by:

\[
m = \frac{y2 - y1}{x2 - x1}
\]

Substituting in our points A (3, 6) and E (12, 9):

\[
m = \frac{9 - 6}{12 - 3} = \frac{3}{9} = \frac{1}{3}
\]

### Step 2: Identify the Response areas:

1. **The slope of the line through AE can be written as**:
- **Response area**: \( \frac{1}{3} \)

2. **This is a Response area relationship because**:
- **Response area**: It shows a linear relationship.

3. **The equation of the line AE is written as**:
- **Response area**: Since slope \( m = \frac{1}{3} \) and we can write it in the slope-intercept form \( y = mx + b \).
- Since it passes through the point (0, 0), it simplifies to:
- **Response area**: \( y = \frac{1}{3}x \)

### Final Statements:

1. The slope of the line through AE can be written as **\(\frac{1}{3}\)**.
2. This is a **linear** relationship because it shows a constant rate of change.
3. The equation of the line AE is written as **\(y = \frac{1}{3}x\)**.

Please note that in the options provided, \(\frac{1}{3}\) is not listed. If using a different point or slope method leads to a different conclusion among the options, please adjust accordingly.